Chapter 12 Resource Masters - Ms. Valeska...The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment

Chapter 12 Resource Masters

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Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish VersionHomework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 12 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition text along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 12 Test Form 2A and Form 2C.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 978-0-07-890521-6MHID: 0-07-890521-4

Printed in the United States of America.

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ContentsTeacher’s Guide to Using the Chapter 12Resource Masters .............................................iv

Chapter ResourcesChapter 12 Student-Built Glossary .................... 1Chapter 12 Anticipation Guide (English) ........... 3Chapter 12 Anticipation Guide (Spanish) .......... 4

Lesson 12-1Representations of Three-Dimensional FiguresStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10Graphing Calculator Activity ............................ 11

Lesson 12-2Surface Area of Prisms and CylindersStudy Guide and Intervention .......................... 12Skills Practice .................................................. 14

Practice ............................................................ 15Word Problem Practice ................................... 16Enrichment ...................................................... 17

Lesson 12-3Surface Area of Pyramids and ConesStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice ............................................................ 21Word Problem Practice ................................... 22Enrichment ...................................................... 23Spreadsheet Activity ........................................ 24

Lesson 12-4Volumes of Prisms and CylindersStudy Guide and Intervention .......................... 25Skills Practice .................................................. 27Practice ............................................................ 28Word Problem Practice ................................... 29Enrichment ...................................................... 30

Lesson 12-5Volumes of Pyramids and ConesStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36

Lesson 12-6Surface Areas and Volumes of SpheresStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42

Lesson 12-7Spherical GeometryStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice ............................................................ 46Word Problem Practice ................................... 47Enrichment ...................................................... 48

Lesson 12-8Congruent and Similar SolidsStudy Guide and Intervention .......................... 49Skills Practice .................................................. 51Practice ............................................................ 52Word Problem Practice ................................... 53Enrichment ...................................................... 54

AssessmentStudent Recording Sheet ................................ 55Rubric for Extended-Response ....................... 56Chapter 12 Quizzes 1 and 2 ........................... 57

Chapter 12 Quizzes 3 and 4 ........................... 58Chapter 12 Mid-Chapter Test .......................... 59Chapter 12 Vocabulary Test ........................... 60Chapter 12 Test, Form 1 ................................. 61Chapter 12 Test, Form 2A ............................... 63Chapter 12 Test, Form 2B ............................... 65Chapter 12 Test, Form 2C .............................. 67Chapter 12 Test, Form 2D .............................. 69Chapter 12 Test, Form 3 ................................. 71Chapter 12 Extended-Response Test ............. 73Standardized Test Practice ............................. 74

Answers ........................................... A1–A36

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iv

Teacher’s Guide to Using the Chapter 12 Resource Masters

The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 12–1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

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Assessment OptionsThe assessment masters in the Chapter 12 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 12 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests• Form 1 contains multiple-choice

questions and is intended for use with approaching grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with beyond grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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NAME DATE PERIOD NAME DATE PERIOD

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Chapter 12 1 Glencoe Geometry

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 12. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter.

Student-Built Glossary12

Vocabulary TermFound

on PageDefi nition/Description/Example

altitude

axis

congruent solids

cross section

Euclidean geometry

great circle

isometric view

lateral area

lateral edge

lateral face

(continued on the next page)

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Vocabulary TermFound

on PageDefi nition/Description/Example

oblique cone

oblique cylinder

oblique prism

regular pyramid

right cone

right cylinder

right prism (PRIZ·uhm)

similar solids

slant height

spherical geometry

Student-Built Glossary (continued)12


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Before you begin Chapter 12

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 12

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

12 Anticipating GuideExtending Surface Area and Volume

Step 1

STEP 1A, D, or NS

StatementSTEP 2A or D

1. The shape of a horizontal cross section of a square pyramid is a triangle.

2. The lateral area of a prism is equal to the sum of the areas of each face.

3. The axis of an oblique cylinder is different than the height of the cylinder.

4. The slant height and height of a regular pyramid are the same.

5. The lateral area of a cone equals the product of π, the radius, and the height of the cone.

6. The volume of a right cylinder with radius r and height h is πr2h.

7. The volume of a pyramid or a cone is found by multiplying the area of the base by the height.

8. To find the surface area of a sphere with radius r, multiply πr2 by 4.

9. All postulates and properties of Euclidean geometry are true in spherical geometry.

10. All spheres and all cubes are similar solids.

Step 2


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NOMBRE FECHA PERÍODO

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Antes de comenzar el Capítulo 12

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Después de completar el Capítulo 12

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

12 Ejercicios PreparationsExtiende el Área de Superficie y volumen

Paso 1

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. La forma de un corte transversal horizontal de una pirámide cuadrada es un triángulo.

2. El área lateral de un prisma es igual a la suma de las áreas de cada cara.

3. El eje de un cilindro oblicuo es diferente a la altura del cilindro.

4. La altura oblicua y la altura de una pirámide regular son las mismas.

5. El área lateral de un cono es igual al producto de π, el radio, por la altura del cono.

6. El volumen de un cilindro recto con radio r y altura h es πr2h.

7. El volumen de una pirámide o un cono se calcula multiplicando el área de la base por la altura.

8. Para calcular el área de superficie de una esfera con radio r, multiplica πr2 por 4.

9. Todos los postulados y propiedades de la geometría euclidiana son verdaderos en geometría esférica.

10. Todas las esferas y todos los cubos son sólidos semejantes.

Paso 2

Capítulo 12 4 Geometría de Glencoe


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Draw Isometric Views Isometric dot paper can be used to draw isometric views, or corner views, of a three-dimensional object on two-dimensional paper.

Use isometric dot paper to sketch a triangular prism 3 units high, with two sides of the base that are 3 unitslong and 4 units long.Step 1 Draw

−−

AB at 3 units and draw −−

AC at 4 units.Step 2 Draw

−−−

AD , −−−

BE , and −−

CF , each at 3 units.Step 3 Draw

−−−

BC and �DEF.

Use isometric dot paper and the orthographic drawing to sketch a solid. • The top view indicates two columns. • The right and left views indicate that the height of figure is

three blocks. • The front view indicates that the columns have heights 2 and 3 blocks.

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of each column.

ExercisesSketch each solid using isometric dot paper.

1. cube with 4 units on each side 2. rectangular prism 1 unit high, 5 units long, and 4 units wide

Use isometric dot paper and each orthographic drawing to sketch a solid.

3.

top view left view front view right view

4.


Study Guide and InterventionRepresentations of Three-Dimensional Figures

12-1

Example 1

A

BC

D

E F

Example 2


object


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Cross Sections The intersection of a solid and a plane is called a cross section of the solid. The shape of a cross section depends upon the angle of the plane.

There are several interesting shapes that are cross sections of a cone. Determine the shape resulting from each cross section of the cone.

a. If the plane is parallel to the base of the cone, then the resulting cross section will be a circle.

b. If the plane cuts through the cone perpendicular to the base and through the center of the cone, then the resulting cross section will be a triangle.

c. If the plane cuts across the entire cone, thenthe resulting cross section will be an ellipse.

ExercisesDescribe each cross section.

1. 2. 3.

Study Guide and Intervention (continued)

Representations of Three-Dimensional Figures

12-1

Example


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12-1

Use isometric dot paper to sketch each prism.

1. cube 2 units on each edge 2. rectangular prism 2 units high, 5 units long, and 2 units wide


3. 4.

Describe each cross section.

5. 6.

7. 8.

Skills PracticeRepresentations of Three-Dimensional Figures

top view left view front view right viewtop view left view front view right view

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Use isometric dot paper to sketch each prism.

1. rectangular prism 3 units high, 2. triangular prism 3 units high, whose bases 3 units long, and 2 units wide are right triangles with legs 2 units and 4 units long


3.


4.


Sketch the cross section from a vertical slice of each figure.

5. 6.

7. SPHERES Consider the sphere in Exercise 5. Based on the cross section resulting from a horizontal and a vertical slice of the sphere, make a conjecture about all spherical cross sections.

8. MINERALS Pyrite, also known as fool’s gold, can form crystals that are perfect cubes. Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectanglar face. What cuts should be made to get each of the shapes? Illustrate your answers.

PracticeRepresentations of Three-Dimensional Figures

12-1


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1. LABELS Jamal removes the label from a cylindrical soup can to earn points for his school. Sketch the shape of the label.

2. BLOCKS Margot’s three-year-old son made the magnetic block sculpture shown below in corner view.

Draw the right view of the sculpture.

3. CUBES Nathan marks the midpoints of three edges of a cube as shown.He then slices the cube along a plane that contains these three points. Describe the resulting cross section.

4. ENGINEERING Stephanie needs an object whose top view is a circle and whose left and front views are squares. Describe an object that will satisfy these conditions.

5. DESK SUPPORTS The figure shows the support for a desk.

a. Draw the top view.

b. Draw the front view.

c. Draw the right view.

Word Problem PracticeRepresentations of Three-Dimensional Figures

12-1


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Drawing Solids on Isometric Dot PaperIsometric dot paper is helpful for drawing solids. Remember to use dashed lines for hidden edges.

For each solid shown, draw another solid whose dimensions are twice as large.

1. 2.

3. 4.

5. 6.

Enrichment12-1


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The science of perspective drawing studies how to draw a three-dimensional object on a two-dimensional page. This science became highly refined during the Renaissance with the work of artists such as Albrecht Dürer and Leonardo da Vinci.

Today, computers are often used to make perspective drawings, particularly elaborate graphics used in television and movies. The three-dimensional coordinates of objects are figured. Then algebra is used to transform these into two-dimensional coordinates. The graph of these new coordinates is called a projection.

The formulas below will draw one type of projection in which the y-axis is drawn horizontally, the z-axis vertically, and the x-axis at an angle of a˚ with the y-axis. If the three-dimensional coordinates of a point are (x, y, z), then the projection coordinates (X, Y) are given by

X = x(-cos a) + y and Y = x(-sin a) + z.Although this type of projection gives a fairly good perspective drawing, it does distort some lengths.

1. The drawing with the coordinates given below is a cube. A(5, 0, 5), B(5, 5, 5), C(5, 5, 0), D(5, 0, 0),

E(0, 0, 5), F(0, 5, 5), G(0, 5, 0), H(0, 0, 0) Use the formulas above to find the projection coordinates of each

point, using a = 45. Round projection coordinates to the nearest integer. Graph the cube on a graphing calculator. Make a sketch of the display.

A'(__, __) B'(__, __) C′(__, __) D′(__, __) E'(__, __) F'(__, __) G′(__, __) H′(__, __)

2. The points A(10, 2, 0), B(10, 10, 0), C(2, 10, 0), and D(3, 3, 4) are vertices of a pyramid. Find the projection coordinates, using a = 25. Round coordinates to the nearest integer. Then graph the pyramid on a graphing calculator by drawing

−−−

A′B′ , −−−

B′C′ , −−−

C′D′ , −−−

D′A′ , and

−−−

D′B′ . Make a sketch of the display. A′(__, __) B′(__, __) C′(__, __) D′(__, __)

Graphing Calculator ActivityPerspective Drawings

12-1

E F

A B

D

H

C

G


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Study Guide and InterventionSurface Areas of Prisms and Cylinders

12-2

Lateral and Surface Areas of Prisms In a solid figure, faces that are not bases are lateral faces. The lateral area is the sum of the area of the lateral faces. The surface area is the sum of the lateral area and the area of the bases.

Lateral Area

of a Prism

If a prism has a lateral area of L square units, a height of h units,

and each base has a perimeter of P units, then L = Ph.

Surface Area

of a Prism

If a prism has a surface area of S square units, a lateral area of L

square units, and each base has an area of B square units, then

S = L + 2B or S = Ph + 2B

Find the lateral and surface area of the regular pentagonal prism above if each base has a perimeter of 75 centimeters and the height is 10 centimeters.

L = Ph Lateral area of a prism

= 75(10) P = 75, h = 10

= 750 Multiply.

The lateral area is 750 square centimeters and the surface area is about 1524.2 square centimeters.

ExercisesFind the lateral area and surface area of each prism. Round to the nearest tenth if necessary.

1.

4 m

3 m10 m

2.

15 in.

10 in.

8 in.

3.

6 in.18 in.

4.

20 cm

10 cm 10 cm

12 cm8 cm9 cm

5.

4 in.

4 in.

12 in.

6.

4 m16 m

Example

pentagonal prism

altitude

lateraledge lateral

face

S = L + 2B = 750 + 2 ( 1 −

2 aP)

= 750 + ( 7.5 − tan 36°

) (75)

≈ 1524.2tan 36° = 7.5 − a

a = 7.5 − tan 36°

a

15 cm

36°


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Lateral and Surface Areas of Cylinders A cylinder is a solid with bases that are congruent circles lying in parallel planes. The axis of a cylinder is the segment with endpoints at the centers of these circles. For a right cylinder, the axis is also the altitude of the cylinder.

Lateral Area

of a Cylinder

If a cylinder has a lateral area of L square units, a height of h units, and a base

has a radius of r units, then L = 2πrh.

Surface Area

of a Cylinder

If a cylinder has a surface area of S square units, a height of h units, and a

base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.

Find the lateral and surface area of the cylinder. Round to the nearest tenth.If d = 12 cm, then r = 6 cm.L = 2πrh Lateral area of a cylinder

= 2π(6)(14) r = 6, h = 14

≈ 527.8 Use a calculator.

S = 2πrh + 2πr2 Surface area of a cylinder

≈ 527.8 + 2π(6)2 2πrh ≈ 527.8, r = 6

≈ 754.0 Use a calculator.

The lateral area is about 527.8 square centimeters and the surface area is about 754.0 square centimeters.

ExercisesFind the lateral area and surface area of each cylinder. Round to the nearest tenth. 1.

12 cm

4 cm 2.

6 in.10 in.

3.

6 cm

3 cm

3 cm

4.

20 cm

8 cm

5.

12 m

4 m

6. 2 m

1 m

radius of baseaxis

base

baseheight

Example

14 cm

12 cm

12-2 Study Guide and Intervention (continued)

Surface Areas of Prisms and Cylinders


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Find the lateral area and surface area of each prism. Round to the nearest tenth if necessary.

1. 2.

3. 4.

Find the lateral area and surface area of each cylinder. Round to the nearest tenth.

5. 6.

7. 8.

12 yd

12 yd

10 yd8 m

6 m

12 m

10 in.5 in.

6 in.

8 in.

9 cm

9 cm

7.8 cm9 cm

12 cm

Skills PracticeSurface Areas of Prisms and Cylinders

12-2

8 in.

12 in.

2 yd

3 yd

12 in.

10 in.

2 m

2 m


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12-2 PracticeSurface Areas of Prisms and Cylinders

Find the lateral and surface area of each prism. Round to the nearest tenth if necessary. 1.

15 cm

15 cm

32 cm

2.

8 ft

10 ft5 ft

3.

2 m11 m

4.

4 yd

4 yd

9.5 yd

5 yd

Find the lateral area and surface area of each cylinder. Round to the nearest tenth.

5. 5 ft

7 ft

6. 4 m

8.5 m

7. 19 in.

17 in.

8.

12 m30 m


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1. LOGOS The Z company specializes in caring for zebras. They want to make a 3-dimensional “Z” to put in front of their company headquarters. The “Z” is 15 inches thick and the perimeter of the base is 390 inches.

15"

What is the lateral surface area of this “Z”?

2. STAIRWELLS Management decides to enclose stairs connecting the first and second floors of a parking garage in a stairwell shaped like an oblique rectangular prism.

16 ft

20 ft

15 ft

9 ft

What is the lateral surface area of the stairwell?

3. CAKES A cake is a rectangular prism with height 4 inches and base 12 inches by 15 inches. Wallace wants to apply frosting to the sides and the top of the cake. What is the surface area of the part of the cake that will have frosting?

4. EXHAUST PIPES An exhaust pipe is shaped like a cylinder with a height of 50 inches and a radius of 2 inches. What is the lateral surface area of the exhaust pipe? Round your answer to the nearest hundredth.

5. TOWERS A circular tower is made by placing one cylinder on top of another. Both cylinders have a height of 18 inches. The top cylinder has a radius of 18 inches and the bottom cylinder has a radius of 36 inches.

18 in.

18 in.

a. What is the total surface area of the tower? Round your answer to the nearest hundredth.

b. Another tower is constructed by placing the original tower on top of another cylinder with a height of 18 inches and a radius of 54 inches. What is the total surface area of the new tower? Round your answer to the nearest hundredth.

12-2 Word Problem PracticeSurface Areas of Prisms and Cylinders


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Enrichment12-2

Minimizing Cost in ManufacturingSuppose that a manufacturer wants to make a can that has a volume of 40 cubic inches. The cost to make the can is 3 cents per square inch for the top and bottom and 1 cent per square inch for the side.

1. Write the value of h in terms of r, given v = πr2h.

2. Write a formula for the cost in terms of r.

3. Use a graphing calculator to graph the formula, letting Y1 represent the cost and X represent r. Use the graph to estimate the point at which the cost is minimized.

4. Repeat the procedure using 2 cents per square inch for the top and bottom and 4 cents per square inch for the top and bottom.

5. What would you expect to happen as the cost of the top and bottom increases?

6. Compute the table for the cost value given. What happens to the height of the can as the cost of the top and bottom increases?

Cost Top Cost Minimum

& Bottom Cylinder h

2 cents 1 cent

3 cents 1 cent

4 cents 1 cent

5 cents 1 cent

6 cents 1 cent

r

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12-3 Study Guide and InterventionSurface Areas of Pyramids and Cones

Lateral and Surface Areas of Pyramids A pyramid is a solid with a polygon base. The lateral faces intersect in a common point known as the vertex. The altitude is the segment from the vertex that is perpendicular to the base. For a regular pyramid, the base is a regular polygon and the altitude has an endpoint at the center of the base. All the lateral edges are congruent and all the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height.

Lateral Area of

a Regular Pyramid

The lateral area L of a regular pyramid is L = 1 − 2 Pℓ, where ℓ

is the slant height and P is the perimeter of the base.

Surface Area of

a Regular Pyramid

The surface area S of a regular pyramid is S = 1 − 2 Pℓ + B,

where ℓ is the slant height, P is the perimeter of the base,

and B is the area of the base.

For the regular square pyramid above, find the lateral area and surface area if the length of a side of the base is 12 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.

Find the slant height.ℓ2 = 62 + 82 Pythagorean Theorem

ℓ2 = 100 Simplify.

ℓ = 10 Take the positive square root of each side.

The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.

ExercisesFind the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1.

15 cm

20 cm

2.

45°

8 ft

3.

60°

10 cm 4.

6 in.8.7 in. 15 in.

lateral edge

baseslant height height

Example

L = 1 − 2 Pℓ Lateral area of a regular pyramid

= 1 − 2 (48)(10) P = 4 � 12 or 48, ℓ = 10

= 240 Simplify.

S = 1 − 2 Pℓ + B Surface area of a regular pyramid

= 240 + 144 1 − 2 Pℓ = 240, B = 12 · 12 or 144

= 384


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Surface Areas of Pyramids and Cones

Lateral and Surface Areas of Cones A cone has a circular base and a vertex. The axis of the cone is the segment with endpoints at the vertex and the center of the base. If the axis is also the altitude, then the cone is a right cone. If the axis is not the altitude, then the cone is an oblique cone.

Lateral Area of

a Cone

The lateral area L of a right circular cone is L = πr�, where r is

the radius and � is the slant height.

Surface Area of

a Cone

The surface area S of a right cone is S = πr� + πr2, where r is

the radius and � is the slant height.

For the right cone above, find the lateral area and surface area if the radius is 6 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.

Find the slant height.ℓ2 = 62 + 82 Pythagorean Theorem

ℓ2 = 100 Simplify.

ℓ = 10 Take the positive square root of each side.

The lateral area is about 188.5 square centimeters and the surface area is about 301.6 square centimeters.

ExercisesFind the lateral area and surface area of each cone. Round to the nearest tenth if necessary.

1.

9 cm

12 cm 2.

5 ft

30°

3. 12 cm

13 cm

4.

4 in.

45°

axis

base base�

slant height

right coneoblique cone

altitudeV V

Example

L = πrℓ Lateral area of a right cone

= π(6)(10) r = 6, ℓ = 10

≈ 188.5 Simplify.

S = πrℓ + πr2 Surface area of a right cone

≈ 188.5 + π(62) πrℓ ≈ 188.5, r = 6

≈ 301.6 Simplify.


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12-3 Skills PracticeSurface Areas of Pyramids and Cones

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

Find the lateral area and surface area of each cone. Round to the nearest tenth.

5. 6.

7. 8.

4 cm

7 cm20 in.

8 in.

9 m

10 m 14 ft

12 ft

14 m

5 m 10 ft

25 ft

8 in.

21 in.

17 mm

9 mm


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12-3 Practice Surface Areas of Pyramids and Cones

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1.

9 yd

10 yd

2. 12 m

7 m

3.

13 ft

5 ft

4.

8 cm

2.5 cm

Find the lateral area and surface area of each cone. Round to the nearest tenth if necessary.

5.

5 m4 m

6. 7 cm

21 cm

7. Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters.

8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.

9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the area of the roof.

10. HATS Cuong bought a conical hat on a recent trip to central Vietnam. The basic frame of the hat is 16 hoops of bamboo that gradually diminish in size. The hat is covered in palm leaves. If the hat has a diameter of 50 centimeters and a slant height of 32 centimeters, what is the lateral area of the conical hat?


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12-3 Word Problem PracticeSurface Areas of Pyramids and Cones

1. PAPER MODELS Patrick is making a paper model of a castle.Part of the model involves cutting out the net shown and folding it into a pyramid. The pyramid has a square base. What is the lateral surface area of the resulting pyramid?

2. TETRAHEDRON Sung Li builds a paper model of a regular tetrahedron, a pyramid with an equilateral triangle for the base and three equilateral triangles for the lateral faces. One of the faces of the tetrahedron has an area of 17 square inches. What is the total surface area of the tetrahedron?

3. PAPERWEIGHTS Daphne uses a paperweight shaped like a pyramid with a regular hexagon for a base. The side length of the regular hexagon is 1 inch. The altitude of the pyramid is 2 inches.

What is the lateral surface area of this pyramid? Round your answers to the nearest hundredth.

4. SPRAY PAINT A can of spray paint shoots out paint in a cone shaped mist. The lateral surface area of the cone is 65π square inches when the can is held 12 inches from a canvas. What is the area of the part of the canvas that gets sprayed with paint? Round your answer to the nearest hundredth.

5. MEGAPHONES A megaphone is formed by taking a cone with a radius of 20 centimeters and an altitude of 60 centimeters and cutting off the tip. The cut is made along a plane that is perpendicular to the axis of the cone and intersects the axis 12 centimeters from the vertex. Round your answers to the nearest hundredth.

a. What is the lateral surface area of the original cone?

b. What is the lateral surface area of the tip that is removed?

c. What is the lateral surface area of the megaphone?

20 cm20 cm15 cm


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Cone Patterns

The pattern at the right is made from a circle. It can be folded to make a cone.

1. Measure the radius of the circle to the nearest centimeter.

2. The pattern is what fraction of the complete circle?

3. What is the circumference of the complete circle?

4. How long is the circular arc that is the outside of the pattern?

5. Cut out the pattern and tape it together to form a cone.

6. Measure the diameter of the circular base of the cone.

7. What is the circumference of the base of the cone?

8. What is the slant height of the cone?

9. Use the Pythagorean Theorem to calculate the height of the cone. Use a decimal approximation. Check your calculation by measuring the height with a metric ruler.

10. Find the lateral area.

11. Find the total surface area.

Make a paper pattern for each cone with the given measurements. Then cut the pattern out and make the cone. Find the measurements.

12.

120°6 cm

13.

20 cm

diameter of base = diameter of base =

lateral area = lateral area =

height of cone = height of cone =(to nearest tenth of a centimeter) (to nearest tenth of a centimeter)

Enrichment12-3


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12-3

You can use a spreadsheet to determine the surface area of a cone.

Lucy wants to wrap a Mother’s Day gift. The gift she has bought for her mother is in a conical box that has a slant height of 6 inches and has a radius of 3 inches. She must determine the surface area of the box to determine how much wrapping paper to buy. Use a spreadsheet to determine the surface area of the box. Round to the nearest tenth.

Step 1 Use cell A1 for the radius of the cone and cell B1 for the height.

Step 2 In cell C1, enter an equals sign followed by PI()*A1*B1 + PI()*A1^2. Then press ENTER. This will return the surface area of the cone.

The surface area of the conical box is 84.8 in2 to the nearest tenth.

Use a spreadsheet to determine the surface area of a cone that has a radius of 2.5 centimeters and a slant height of 5.2 centimeters. Round to the nearest tenth.

Step 1 Use cell A2 for the radius of the cone and cell B2 for the slant height.

Step 2 Click on the bottom right corner of cell C1 and drag it to C2. This returns the surface area of the cone.

The surface area of the cone is 60.5 cm2 to the nearest tenth.

ExercisesUse a spreadsheet to find the surface area of each cone with the given dimensions. Round to the nearest tenth.

1. r = 12 m, � = 2.3 m 2. r = 6 m, � = 2 m

3. r = 3 in., � = 7 in. 4. r = 5 in., � = 11 in.

5. r = 1 ft, � = 3 ft 6. r = 3 ft, � = 1.5 ft

7. r = 10 mm, � = 20 mm 8. r = 1.5 mm, � = 4.5 mm

9. r = 6.2 cm, � = 1.2 cm 10. r = 10 cm, � = 15 cm

11. r = 10 m, � = 2 m 12. r = 11 m, � = 13 m

Spreadsheet ActivitySurface Areas of Cones

Example 1

Example 2

A12

B C

Sheet 1


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Study Guide and InterventionVolumes of Prisms and Cylinders

Volumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.

Volume

of a Prism

If a prism has a volume of V cubic units, a height of h units,

and each base has an area of B square units, then V = Bh.

Find the volume of the prism.

7 cm3 cm

4 cm

V = Bh Volume of a prism

= (7)(3)(4) B = (7)(3), h = 4

= 84 Multiply.

The volume of the prism is 84 cubic centimeters.

Find the volume of the prism if the area of each base is 6.3 square feet.

3.5 ft

base

V = Bh Volume of a prism

= (6.3)(3.5) B = 6.3, h = 3.5

= 22.05 Multiply.The volume is 22.05 cubic feet.

Exercises

Find the volume of each prism.

1.

8 ft

8 ft

8 ft

2.

3 cm

4 cm

1.5 cm

3.

30°15 ft

12 ft 4.

10 ft15 ft

12 ft

5.

4 cm

6 cm

2 cm

1.5 cm

6.

7 yd4 yd

3 yd

12-4

Example 1 Example 2

cubic foot cubic yard27 cubic feet = 1 cubic yard


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Volumes of Prisms and Cylinders

Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. When a solid is not a right solid, use Cavalieri’s Priniciple to find the volume. The principle states that if two solids have the same height and the same cross sectional area at every level, then they have the same volume.

Find the volume of the cylinder.

4 cm

3 cm

V = πr2h Volume of a cylinder

= π(3)2(4) r = 3, h = 4

≈ 113.1 Simplify.

The volume is about 113.1 cubic centimeters.

Find the volume of the oblique cylinder.

8 in.

13 in.

5 in.

h

Use the Pythagorean Theorem to find the height of the cylinder.h2 + 52 = 132 Pythagorean Theorem

h2 = 144 Simplify.

h = 12 Take the positive square root of each side.

V = πr2h Volume of a cylinder

= π(4)2(12) r = 4, h = 12

≈ 603.2 Simplify.

The Volume is about 603.2 cubic inches.ExercisesFind the volume of each cylinder. Round to the nearest tenth.

1. 2 ft

1 ft

2. 18 cm

2 cm

3.

12 ft1.5 ft

4. 20 ft

20 ft

5.

10 cm

13 cm

6.

1 yd4 yd

12-4

Volume of

a Cylinder

If a cylinder has a volume of V cubic units, a height of h units,

and the bases have a radius of r units, then V = πr 2h.

Example 1 Example 2

r

h


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12-4

Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1.

18 cm

16 cm

8 cm 2.

6 ft

8 ft

2 ft

3.

3 m

5 m

13 m

4.

16 in. 22 in.

34 in.

5.

15 mm23 mm

6. 6 yd

10 yd

Find the volume of each oblique prism or cylinder. Round to the nearest tenth if necessary.

7. 8.

5 in.

3 in.

Skills PracticeVolumes of Prisms and Cylinders

17 cm

18 cm

4 cm


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PracticeVolumes of Prisms and Cylinders

Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1.

17 m10 m

26 m 2.

5 in.

5 in.

5 in.

9 in.

3.

16 mm 17.5 mm

4. 7 ft 25 ft

5.

13 yd

20 yd

10 yd 6.

30 cm

8 cm

7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25 1 −

2 inches and a radius of 21 inches.

a. What is the volume of the aquarium in cubic feet?

b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold?

c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in the aquarium to the nearest five pounds?

12-4


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Word Problem PracticeVolumes of Prisms and Cylinders

1. TRASH CANS The Meyer family uses a kitchen trash can shaped like a cylinder. It has a height of 18 inches and a base diameter of 12 inches.What is the volume of the trash can? Round your answer to the nearest tenth of a cubic inch.

2. BENCH Inside a lobby, there is a piece of furniture for sitting. The furniture is shaped like a simple block with a square base 6 feet on each side and a height of 1 3−

5 feet.

6 ft6 ft

1 ft35

What is the volume of the seat?

3. FRAMES Margaret makes a square frame out of four pieces of wood. Each piece of wood is a rectangular prism with a length of 40 centimeters, a height of 4 centimeters, and a depth of 6 centimeters. What is the total volume of the wood used in the frame?

4. PENCIL GRIPS A pencil grip is shaped like a triangular prism with a cylinder removed from the middle. The base of the prism is a right isosceles triangle with leg lengths of 2 centimeters. The diameter of the base of the removed cylinder is 1 centimeter. The heights of the prism and the cylinder are the same, and equal to 4 centimeters.

What is the exact volume of the pencil grip?

5. TUNNELS Construction workers are digging a tunnel through a mountain. The space inside the tunnel is going to be shaped like a rectangular prism. The mouth of the tunnel will be a rectangle 20 feet high and 50 feet wide and the length of the tunnel will be 900 feet.

a. What will the volume of the tunnel be?

b. If instead of a rectangular shape, the tunnel had a semicircular shape with a 50-foot diameter, what would be its volume? Round your answer to the nearest cubic foot.

18 in.

12 in.

12-4


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Visible Surface AreaUse paper, scissors, and tape to make five cubes that have one-inch edges. Arrange the cubes to form each shape shown. Then find the volume and the visible surface area. In other words, do not include the area of surface covered by other cubes or by the table or desk.

1. 2.

volume = volume =

visible surface area = visible surface area =

3. 4. 5.

volume = volume = volume =

visible surface area = visible surface area = visible surface area =

6. Find the volume and the visible surface area of the figure at the right.

volume =

visible surface area =

4 in.

4 in.

3 in.

8 in.

3 in.

5 in.

5 in.

3 in.

3 in.

12-4 Enrichment


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Study Guide and InterventionVolumes of Pyramids and Cones

Volumes of Pyramids This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically, the volume of the pyramid is one-third of the volume of the prism.

Find the volume of the square pyramid.

V = 1 − 3 Bh Volume of a pyramid

= 1 − 3 (8)(8)10 B = (8)(8), h = 10

≈ 213.3 Multiply.

The volume is about 213.3 cubic feet.

ExercisesFind the volume of each pyramid. Round to the nearest tenth if necessary.

1.

12 ft

8 ft

10 ft 2.

10 ft

6 ft15 ft

3.

4 cm8 cm

12 cm 4.

18 ft

regularhexagon 6 ft

5.

15 in.

15 in.

16 in. 6. 6 yd

8 yd

5 yd

8 ft

8 ft

10 ft

12-5

Volume of

a Pyramid

If a pyramid has a volume of V cubic units, a height of h units,

and a base with an area of B square units, then V = 1 − 3 Bh.

Example


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Volumes of Cones For a cone, the volume is one-third the product of the height and the area of the base. The base of a cone is a circle, so the area of the base is πr2.

Find the volume of the cone.

V = 1 − 3 πr2h Volume of a cone

= 1 − 3 π(5)212 r = 5, h = 12

≈ 314.2 Simplify.

The volume of the cone is about 314.2 cubic centimeters.

ExercisesFind the volume of each cone. Round to the nearest tenth.

1.

6 cm10 cm

2. 8 ft

10 ft

3.

30 in.

12 in.

4. 45°

18 yd

20 yd

5. 26 ft

20 ft

6.

16 cm

45°

12 cm

5 cm


Volumes of Pyramids and Cones

12-5

Volume of

a Cone

If a cone has a volume of V cubic units, a height of h units,

and the bases have a radius of r units, then V = 1 − 3 πr2h.

Example

r

h


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12-5 Skills PracticeVolumes of Pyramids and Cones

Find the volume of each pyramid or cone. Round to the nearest tenth if necessary. 1. 2.

3. 4.

5. 6.

Find the volume of each oblique pyramid or cone. Round to the nearest tenth if necessary.

7. 8.

5 ft5 ft

8 ft

4 cm7 cm

8 cm

8 in.10 in.

14 in.25 m

12 m

25 yd

14 yd66°

18 mm

4 ft4 ft

6 ft12 cm

6 cm


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PracticeVolumes of Pyramids and Cones

Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1.

9.2 yd9.2 yd

13 yd

2.

12.5 cm25 cm

23 cm

3.

19 ft

9 ft 4.

52°

12 mm

5.

6 in.6 in.

11 in.

6.

37 ft11 ft

7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of the shed?

8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the 27th century B.C. In its original state, it stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.

12-5


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1. ICE CREAM DISHES The part of a dish designed for ice cream is shaped like an upside-down cone. The base of the cone has a radius of 2 inches and the height is 1.2 inches.

What is the volume of the cone? Round your answer to the nearest hundredth.

2. GREENHOUSES A greenhouse has the shape of a square pyramid. The base has a side length of 30 yards. The height of the greenhouse is 18 yards.

18

yd

30 yd

What is the volume of the greenhouse?

3. TEEPEE Caitlyn made a teepee for a class project. Her teepee had a diameter of 6 feet. The angle the side of the teepee made with the ground was 65°.

65˚

What was the volume of the teepee? Round your answer to the nearest hundredth.

4. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the base of the cone and cylinder is 2 feet.

What is the volume of the stone that the sculptor must remove? Round your answer to the nearest hundredth.

5. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet.

12 feet

16 feet

3 feet

a. What is the volume of the entire square pyramid that the stage is part of?

b. What is the volume of the top of the pyramid that is removed to get the stage?

c. What is the volume of the stage?

Word Problem PracticeVolumes of Pyramids and Cones

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Enrichment

FrustumsA frustum is a figure formed when a plane intersects a pyramid or cone so that the plane is parallel to the solid’s base. The frustum is the part of the solid between the plane and the base. To find the volume of a frustum, the areas of both bases must be calculated and used in the formula.

V = 1 − 3 h(B1 + B2 + √ �� B1B2 ),

where h = height (perpendicular distance between the bases),B1 = area of top base, and B2 = area of bottom base.

Describe the shape of the bases of each frustum. Then find the volume. Round to the nearest tenth.

1. 13 cm

6 cm

9 cm

5 cm

19.5 cm

2.

7.5 in.

4.5 in.

3 in.

3.

8 m

6 m

12 m

4.5 m2.25 m

3 m

5 m

4.

12 ft13 ft

7 ft

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Study Guide and InterventionSurface Areas and Volumes of Spheres

Surface Areas of Spheres You can think of the surface area of a sphere as the total area of all of the nonoverlapping strips it would take to cover the sphere. If r is the radius of the sphere, then the area of a great circle of the sphere is πr2. The total surface area of the sphere is four times the area of a great circle.

Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 centimeters.

S = 4πr2 Surface area of a sphere

= 4π(6)2 r = 6

≈ 452.4 Simplify.

The surface area is 452.4 square centimeters.

ExercisesFind the surface area of each sphere or hemisphere. Round to the nearest tenth.

1. 5 m

2.

7 in

3.

3 ft

4.

9 cm

5. sphere: circumference of great circle = π cm

6. hemisphere: area of great circle ≈ 4π ft2

r

6 cm

Surface Area

of a SphereIf a sphere has a surface area of S square units and a radius of r units, then S = 4πr2.

Example

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Surface Areas and Volumes of Spheres

12-6

Volumes of Spheres A sphere has one basic measurement, the length of its radius. If you know the length of the radius of a sphere, you can calculate its volume.

Find the volume of a sphere with radius 8 centimeters.

V = 4 − 3 πr3 Volume of a sphere

= 4 − 3 π (8)3 r = 8

≈ 2144.7 Simplify.

The volume is about 2144.7 cubic centimeters.

ExercisesFind the volume of each sphere or hemisphere. Round to the nearest tenth.

1.

5 ft

2. 6 in. 3.

16 in.

4. hemisphere: radius 5 in.

5. sphere: circumference of great circle ≈ 25 ft

6. hemisphere: area of great circle ≈ 50 m2

r

8 cm

Volume of

a SphereIf a sphere has a volume of V cubic units and a radius of r units, then V = 4 −

3 πr3.

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Skills PracticeSurface Areas and Volumes of Spheres

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

1.

7 in.

2.

32 m

3. hemisphere: radius of great circle = 8 yd

4. sphere: area of great circle ≈ 28.6 in2

Find the volume of each sphere or hemisphere. Round to the nearest tenth.

5.

16.2 cm

6.

94.8 ft

7. hemisphere: diameter = 48 yd

8. sphere: circumference of a great circle ≈ 26 m

9. sphere: diameter = 10 in.

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12-6 PracticeSurface Areas and Volumes of Spheres

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

1.

6.5 cm

2.

89 ft

3. hemisphere: radius of great circle = 8.4 in.

4. sphere: area of great circle ≈ 29.8 m2

Find the volume of each sphere or hemisphere. Round to the nearest tenth.

5.

12.32 ft

6.32 m

7. hemisphere: diameter = 18 mm

8. sphere: circumference ≈ 36 yd

9. sphere: radius = 12.4 in.


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Word Problem PracticeSurface Areas and Volumes of Spheres

1. ORANGES Mandy cuts a spherical orange in half along a great circle. If the radius of the orange is 2 inches, what is the area of the cross section that Mandy cut? Round your answer to the nearest hundredth.

2. BILLIARDS A billiard ball set consists of 16 spheres, each 2 1

−4

inches in diameter. What is the total volume of a complete set of billiard balls? Round your answer to the nearest thousandth of a cubic inch.

3. MOONS OF SATURN The planet Saturn has several moons. These can be modeled accurately by spheres. Saturn’s largest moon Titan has a radius of about 2575 kilometers. What is the approximate surface area of Titan? Round your answer to the nearest tenth.

4. THE ATMOSPHERE About 99% of Earth’s atmosphere is contained in a 31-kilometer thick layer that enwraps the planet. The Earth itself is almost a sphere with radius 6378 kilometers. What is the ratio of the volume of the atmosphere to the volume of Earth? Round your answer to the nearest thousandth.

5. CUBES Marcus builds a sphere inside ofa cube. The sphere fits snugly inside the cube so that the sphere touches the cube at one point on each side. The side length of the cube is 2 inches.

a. What is the surface area of the cube?

b. What is the surface area of the sphere? Round your answers to the nearest hundredth.

c. What is the ratio of the surface area of the cube to the surface area of the sphere? Round your answer to the nearest hundredth.

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Enrichment

Spheres and DensityThe density of a metal is a ratio of its mass to its volume. For example, the mass of aluminum is 2.7 grams per cubic centimeter. Here is a list of several metals and their densities.

Aluminum 2.7 g/cm3 Copper 8.96 g/cm3

Gold 19.32 g/cm3 Iron 7.874 g/cm3

Lead 11.35 g/cm3 Platinum 21.45 g/cm3

Silver 10.50 g/cm3

To calculate the mass of a piece of metal, multiply volume by density.

Find the mass of a silver ball that is 0.8 cm in diameter.

M = D · V

= 10.5 · 4 − 3 π(0.4)3

≈ 10.5(0.27)≈ 2.81

The mass is about 2.81 grams.

ExercisesFind the mass of each metal ball described. Assume the balls are spherical. Round your answers to the nearest tenth.

1. a copper ball 1.2 cm in diameter

2. a gold ball 0.6 cm in diameter

3. an aluminum ball with radius 3 cm

4. a platinum ball with radius 0.7 cm

Solve. Assume the balls are spherical. Round your answers to the nearest tenth.

5. A lead ball weighs 326 grams. Find the radius of the ball to the nearest tenth of a centimeter.

6. An iron ball weighs 804 grams. Find the diameter of the ball to the nearest tenth of a centimeter.

7. A silver ball and a copper ball each have a diameter of 3.5 centimeters. Which weighs more? How much more?

8. An aluminum ball and a lead ball each have a radius of 1.2 centimeters. Which weighs more? How much more?

12-6

Example


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12-7 Study Guide and InterventionSpherical Geometry

Geometry On A Sphere Up to now, we have been studying Euclidean geometry, where a plane is a flat surface made up of points that extends infinitely in all directions. In spherical geometry, a plane is the surface of a sphere.

Name each of the following on sphere K.

a. two lines containing the point F

� �� EG and � �� BH are lines on sphere K that contain the point F

b. a line segment containing the point J

−−

ID is a segment on sphere K that contains the point J

c. a triangle

�AHI is a triangle on sphere K

ExercisesName two lines containing point Z, a segment containing point R, and a triangle in each of the following spheres.

1.

F

2.

M

Determine whether figure u on each of the spheres shown is a line in spherical geometry.

3. 4.

5. GEOGRAPHY Lines of latitude run horizontally across the surface of Earth. Are there any lines of latitude that are great circles? Explain.

Example

K


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Comparing Euclidean and Spherical Geometries Some postulates and properties of Euclidean geometry are true in spherical geometry. Others are not true or are true only under certain circumstances.

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

Given any line, there are an infinite number of parallel lines.

On the sphere to the right, if we are given line m we see that it goes through the poles of the sphere. If we try to make any other line on the sphere, it would intersect line m at exactly 2 points. This property is not true in spherical geometry.A corresponding statement in spherical geometry would be: “Given any line, there are no parallel lines.”

ExercisesTell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

1. If two nonidentical lines intersect at a point, they do not intersect again.

2. Given a line and a point on the line, there is only one perpendicular line going through that point.

3. Given two parallel lines and a transversal, alternate interior angles are congruent.

4. If two lines are perpendicular to a third line, they are parallel.

5. Three noncollinear points determine a triangle.

6. A largest angle of a triangle is opposite the largest side.


Spherical Geometry

Example


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Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.

1.

C

2.

L


3. 4.

basketball

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

5. If two lines form vertical angles, then the angles are equal in measure.

6. If two lines meet a third line at the same angle, those lines are parallel.

7. Two lines meet at two 90° angles or they meet at angles whose sum is 180°.

8. Three non-parallel lines divide the plane into 7 separate parts.

Skills PracticeSpherical Geometry

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Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.

1.

L

2.

M


3.

tennis ball

4.

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. 5. A triangle can have at most one obtuse angle.

6. The sum of the angles of a triangle is 180°.

7. Given a line and a point not on the line, there is exactly one line that goes through the point and is perpendicular to the line.

8. All equilateral triangles are similar.

9. AIRPLANES When flying an airplane from New York to Seattle, what is the shortest route: flying directly west, or flying north across Canada? Explain.

12-7 PracticeSpherical Geometry


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12-7 Word Problem PracticeSpherical Geometry

1. PAINTING Consider painting quadrilateral ABCD on the beach ball with radius 1 ft. What is the surface area you would need to paint?

2. EARTH The Equator and the Prime Meridian are perpendicular great circles that divide Earth into North, South and East, West hemispheres. If Earth has a surface area of 197,000,000 square miles, what is the surface area of the North-East section of Earth?

Source: NASA

3. OCEAN If the oceans cover 70% of Earth’s surface, what is the surface area of the oceans?

Source: NASA

4. GEOMETRY Three nonidentical lines on the circle divide it into either 6 sections or 8 triangles. What condition is needed so that the three lines form 6 sections?

5. GEOGRAPHY Latitude and longitude lines are imaginary lines on Earth. The lines of latitude are horizontal concentric circles that help to define the distance a place is from the equator. Lines of latitude are measured in degrees. The equator is 0°. The north pole is 90° north latitude. The lines of longitude are great circles that help to define the distance a place is from the Prime Meridan, which is located in England and considered the longitude of 0°.

a. The mean radius of Earth is 3963 miles. Atlanta, Georgia, has coordinates (33°N, 84°W) and Cincinnati, Ohio, has coordinates (39°N, 84°W). Estimate the distance between Atlanta and Cincinnati to the nearest tenth.

b. Seattle, Washington, has coordinates (47°N, 122°W) and Portland, Oregon,has coordinates (45°N, 122°W). Estimate the distance between Portland and Seattle to the nearest tenth.

A

C

D B


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12-7

ProjectionsWhen making maps of Earth, cartographers must show a sphere on a plane. To do this they have to use projections, a method of converting a sphere into a plane. But these projections have their limitations.

The map on the right is a Mercator projection of Earth. On this map Greenland appears to be the same size as Africa. But Greenland has a land area of 2,166,086 square kilometers and Africa has a land area of 30,365,700 square kilometers.

The map on the right is a Lambert projection. When a pilot draws a straight line between two points on this map the line shows true bearing, or relative direction to the North Pole. However, the bottom area of this map distorts distances.

1. When would it be useful to use a Mercator projection of Earth?

2. Does each square on the Mercator projection have the same surface area? Explain.

3. Does each square on the Lambert projection have the same surface area? Explain.

4. The Mercator projection uses a cylinder to map Earth, while the Lambert projection uses a cone to map Earth. What other shapes do you think could be used to map Earth?

EnrichmentSpherical Geometry

60˚ N

180˚ W 180˚ E90˚ W 0˚ 90˚ E

40˚ N20˚ N0˚

20˚ S40˚ S60˚ S


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Identify Congruent or Similar Solids Similar solids have exactly the same shape but not necessarily the same size. Two solids are similar if they are the same shape and the ratios of their corresponding linear measures are equal. All spheres are similar and all cubes are similar. Congruent solids have exactly the same shape and the same size. Congruent solids are similar solids with a scale factor of 1:1. Congruent solids have the following characteristics:

• Corresponding angles are congruent• Corresponding edges are congruent• Corresponding faces are congruent• Volumes are equal

Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

ratio of width: 3 − 6 = 1 −

2 ratio of length: 4 −

8 = 1 −

2

ratio of hypotenuse: 5 − 10

= 1 − 2 ratio of height: 4 −

8 = 1 −

2

The ratios of the corresponding sides are equal, so the triangular prisms are similar. The scale factor is 1:2. Since the scale factor is not 1:1, the solids are not congruent.

ExercisesDetermine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

1.

2 cm

1 cm

10 cm

5 cm

2.

4.2 in. 12.3 in.

4.2 in.

12.3 in.

3.

4 in.

8 in.

4.

2 m

2 m

4 m1 m

1 m3 m

Study Guide and InterventionCongruent and Similar Solids

12-8

5 in.

4 in.

4 in.3 in.

10 in.

8 in.

8 in.

6 in.Example


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Properties of Congruent or Similar Solids When pairs of solids are congruent or similar, certain properties are known.

If two similar solids have a scale factor of a:b then,• the ratio of their surface areas is a2:b2.• the ratio of their volumes is a3:b3.

Two spheres have radii of 2 feet and 6 feet. What is the ratio of the volume of the small sphere to the volume of the large sphere?

First, find the scale factor.

radius of the small sphere

−− radius of the large sphere

= 2 − 6 or 1 −

3

The scale factor is 1 − 3 .

a3 −

b3 = (1)3

− (3)3

or 1 − 27

So, the ratio of the volumes is 1:27.

Exercises 1. Two cubes have side lengths of 3 inches and 8 inches. What is the ratio of the surface

area of the small cube to the surface area of the large cube?

2. Two similar cones have heights of 3 feet and 12 feet. What is the ratio of the volume of the small cone to the volume of the large cone?

3. Two similar triangular prisms have volumes of 27 square meters and 64 square meters. What is the ratio of the surface area of the small prism to the surface area of the large prism?

4. COMPUTERS A small rectangular laptop has a width of 10 inches and an area of 80 square inches. A larger and similar laptop has a width of 15 inches. What is the length of the larger laptop?

5. CONSTRUCTION A building company uses two similar sizes of pipes. The smaller size has a radius of 1 inch and length of 8 inches. The larger size has a radius of 2.5 inches What is the volume of the larger pipes?


Congruent and Similar Solids

Example

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2 ft

6 ft


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Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

1.

2 cm

4 cm

3 cm

6 cm

12 cm

9 cm 2.

6 cm

9 cm

8 cm

12 cm

3.

5 m 10 m

4.

3 ft

1 ft1 ft 9 ft

3 ft3 ft

5. Two similar pyramids have heights of 4 inches and 7 inches What is the ratio of the volume of the small pyramid to the volume of the large pyramid?

6. Two similar cylinders have surface areas of 40π square feet and 90π square feet. What is the ratio of the height of the large cylinder to the height of the small cylinder?

7. COOKING Two stockpots are similar cylinders. The smaller stockpot has a height of 10 inches and a radius of 2.5 inches. The larger stockpot has a height of 16 inches. What is the volume of the larger stockpot? Round to the nearest tenth.

Skills PracticeCongruent and Similar Solids

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Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

1.

6 cm

8 cm

18 cm

24 cm

2.

5 cm

12 cm

10 cm

24 cm

3. 1 m 1 m5 m 5 m

3 m 3 m

4 m4 m

4.

5 cm

5 cm

2 cm

10 cm

10 cm

1.5 cm

5. Two cubes have surface areas of 72 square feet and 98 square feet. What is the ratio of the volume of the small cube to the volume of the large cube?

6. Two similar ice cream cones are made of a half sphere on top and a cone on bottom. They have radii of 1 inch and 1.75 inches respectively. What is the ratio of the volume of the small ice cream cone to the volume of the large ice cream cone? Round to the nearest tenth.

7. ARCITHECTURE Architects make scale models of buildings to present their ideas to clients. If an architect wants to make a 1:50 scale model of a 4000 square foot house, how many square feet will the model have?

12-8 PracticeCongruent and Similar Solids


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12-8 Word Problem PracticeCongruent and Similar Solids

1. COOKING A cylindrical pot is 4.5 inchestall and has a radius of 4 inches. How tall would a similar pot be if its radius is 6 inches?

2. MANUFACTURING Boxes, Inc. wants to make the two boxes below. How long does the second box need to be so that they are similar?

24 cm

15 cm 25 cm

25 cm15 cm

3. FARMING A farmer has two similar cylindrical grain silos. The smaller silo is 25 feet tall and the larger silo is 40 feet tall. If the smaller silo can hold 1500 cubic feet of grain, how much can the larger silo hold?

4. PLANETS Earth has a surface area of about 196,937,500 square miles. Mars has a surface area of about 89,500,000 square miles. What is the ratio of the radius of Earth to the radius of Mars? Round to the nearest tenth.

Source: NASA

5. BASEBALL Major League Baseball or MLB, rules state that baseballs must have a circumference of 9 inches. The National Softball Association, or NSA, rules state that softballs must have a circumference not exceeding 12 inches.

Source: MLB, NSA

a. Find the ratio of the circumference of MLB baseballs to the circumference of NSA softballs.

b. Find the ratio of the volume of MLB baseballs to the volume of NSA softballs. Round to the nearest tenth.


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Enrichment

Doubling SizesConsider what happens to surface area when the sides of a figure are doubled.

The sides of the large cube are twice the size of the sides of the small cube.

1. How long are the edges of the large cube?

2. What is the surface area of the small cube?

3. What is the surface area of the large cube?

4. The surface area of the large cube is how many times greater than that of the small cube?

The radius of the large sphere at the right is twice the radius of the small sphere.

5. What is the surface area of the small sphere?

6. What is the surface area of the large sphere?

7. The surface area of the large sphere is how many times greater than the surface area of the small sphere?

8. It appears that if the dimensions of a solid are doubled, the surface area is multiplied by .

Now consider how doubling the dimensions affects the volume of a cube.

The sides of the large cube are twice the size of the sides of the small cube.

9. How long are the edges of the large cube?

10. What is the volume of the small cube?

11. What is the volume of the large cube?

12. The volume of the large cube is how many times greater than that of the small cube?

The large sphere at the right has twice the radius of the small sphere.

13. What is the volume of the small sphere?

14. What is the volume of the large sphere?

15. The volume of the large sphere is how many times greater than the volume of the small sphere?

16. It appears that if the dimensions of a solid are doubled, the volume is multiplied by .

5 in.

3 m

5 in.

3 m

12-8


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12 Student Recording SheetUse this recording sheet with pages 894– 895 of the Student Edition.

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

Multiple Choice

Extended Response

Record your answers for Question 13 on the back of this paper.

9.

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.

7. ————————

8. ————————

9. ———————— (grid in)

10. ————————

11. ———————— (grid in)

12. ————————

Short Response/Gridded Response

9

8

7

6

5

4

3

2

1

0

9

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12

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show how he or she arrived at the answer, the student will be awarded only 1 credit. All extended response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.

• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.

Exercise 13 Rubric

Score Specific Criteria

4 A correct solution that is supported by well-developed, accurate explanations. The scale factor of the prisms, and the correct volumes of the prisms are provided. The student displays an understanding of how volume changes based upon changing dimensions by correctly answering parts c and d.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.

Rubric for Scoring Extended-Response

056-065_GEOCRMC12_890521.indd 56056-065_GEOCRMC12_890521.indd 56 5/30/09 3:11:24 PM5/30/09 3:11:24 PM

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12

12

1. Given the corner view of a figure, draw the left view.

2. A cylinder has a lateral area of 120π square meters, and a height of 7 meters. Find the radius. Round to the nearest tenth.

3. Find the lateral area of the hexagonal prism.

4. Find the surface area of a rectangular prism with a length and width of 6 centimeters and a height of 12 centimeters.

5. MULTIPLE CHOICE Find the surface area of the prism to the nearest hundredth.

A 30.50 B 54.00

C 49.45 D 52.44

1.

2.

3.

4.

5.

1. Find the surface area of the solid figureat the right to the nearest tenth.

For Questions 2 and 3, use a right circular cone with a radius of 5 feet and a slant height of 12 feet. Round to the nearest tenth.


3. Find the surface area.

4. A rectangular prism has a length of 16 feet, a width of 9 feet, and a height of 8 feet. Find the volume of the prism.

5. A cylinder has a diameter of 20 inches and a height of 9 inches. Find the volume of the cylinder, round to the nearest tenth.

Chapter 12 Quiz 1(Lessons 12-1 and 12-2)


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12 Chapter 12 Quiz 3(Lessons 12-5 and 12-6)

1. A pyramid has a height of 18 centimeters and a base with an area of 26 square centimeters. Find the volume.

2. Find the volume of the cone. Round to the nearest tenth.

3. A hemisphere has a base with an area that is 25π square centimeters. Find the volume of the hemisphere. Round to the nearest tenth.

4. A sphere has a great circle with a circumference of 8π meters. What is the surface area of the sphere?

5. MULTIPLE CHOICE A sphere has a radius that is 15.6 inches long. Find the volume of the sphere. Round to the nearest tenth.

A 1019.4 in3 C 15,902 in3

B 7951.2 in3 D 47,702.2 in3

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2.

3.

4.

5.

1. Name two lines containing point Z, a segment containing point R, and a triangle in the sphere F.

2. Do all lines have an infinite number of points in spherical geometry? If not, explain your reasoning.

3. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

4. Two similar prisms have heights of 12 feet and 20 feet. What is the ratio of the volume of the small prism to the volume of the large prism?

5. Two cubes have surface areas of 81 square inches and 144 square inches. What is the ratio of the volume of the small cube to the volume of the large cube?

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12

16 cm

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1 m 0.5 m


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12 Chapter 12 Mid-Chapter Test(Lessons 12-1 through 12-4)

1.

2.

3.

4.

5.

1. A cylinder is standing on one of its bases. It is sliced by a plane horizontally. What is the shape of the cross section?

A triangle C circle B square D rectangle

2. Choose the correct formula for the surface area of a cone. F S = Ph + 2B H S = 1 −

2 P� + B

G S = πr� + πr2 J S = πr� + 2πr

3. The surface area of a prism is 120 square centimeters and the area of each base is 32 square centimeters. Find the lateral area of the prism.A 184 cm2 B 152 cm2 C 86 cm2 D 56 cm2

For Questions 4 and 5, refer to the solid figure. Round to the nearest tenth. 4. Find the lateral area.

F 9289.1 ft2 H 10,965.4 ft2

G 9434.2 ft2 J 12,641.8 ft2

5. Find the surface area.A 9289.1 ft2 B 9434.2 ft2 C 10,965.4 ft2 D 12,641.8 ft2

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Part II

6. Draw the top view of this orthogonal drawing.

For Questions 7 and 8, refer to the regular hexagonal prism.

7. Sketch the cross section from a vertical slice of the figure.

8. Find the surface area. Round to the nearest tenth.

9. A barrel in the shape of a right cylinder has a diameter of 18 inches and a height of 42 inches. Find the surface area of the barrel.

10. Find the lateral area of the solid. Round to the nearest tenth.

Part I Write the letter for the correct answer in the blank at the right of each question.

4 in.

6 in.

2 6

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12 Chapter 12 Vocabulary Test

altitude

axis

base edges

composite solid

congruent solid

cross section

Euclidean geometry

great circle

isometric view

lateral area

lateral edge

lateral face

non-Euclidean geometry

oblique cone

oblique cylinder

oblique prism

regular pyramidright cone

right cylinder

right prism

similar solids

slant height

spherical geometry

topographical map

Choose from the terms above to complete each sentence.

1. The height of each lateral face of a regular pyramid is called a(n) .

2. have the same shape but not the same size.

3. If the axis of a cylindrical is also the altitude, then the cylinder is called a(n) .

4. is the measure of the amount of space that a figure encloses.

Choose the correct term to complete each sentence.

5. The segment whose endpoints are the centers of the circular bases of a cylinder is the (axis, hemisphere)

6. Two solids that have the same shape and the same size are called (congruent solids, composite solids).

State whether each sentence is true or false. If false, replace the underlined word or phrase to make a true sentence.

7. A polyhedron that has all but one face intersecting at one point is a prism.

8. A cross section is the intersection of a plane and a solid figure.

9. A hexagonal prism has a six lateral faces.

10. A cone with an axis that is not an altitude is a right cone.

Define each term in your own words.

11. lateral area

12. right prism

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12.

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12 Chapter 12 Test, Form 1

Write the letter for the correct answer in the blank at the right of each question.

1. Which of these is part of an orthographic drawing? A a perspective view C a two-dimensional top view B a corner view D a three-dimensional view

For Questions 2–4, refer to the figure.

2. Identify this solid figure. F square pyramid H triangular pyramid G square prism J triangular prism

3. Name the base. A �ABE B �ABCD C �CDE D E

4. The shape of a vertical cross section of a cone is a . F circle G triangle H square J trapezoid

5. Find the surface area of the cube. A 9 in2 C 36 in2

B 27 in2 D 54 in2

For Questions 6 and 7, refer to the figure.

6. Find the lateral area. Round to the nearest tenth. F 75.4 ft2 H 50.3 ft2

G 62.8 ft2 J 25.1 ft2

7. Find the surface area. Round to the nearest tenth. A 75.4 ft2 B 62.8 ft2 C 50.3 ft2 D 25.1 ft2

For Questions 8 and 9, refer to the figure.

8. Find the lateral area. F 108 cm2 H 162 cm2

G 144 cm2 J 324 cm2

9. Find the surface area. A 108 cm2 B 144 cm2 C 162 cm2 D 324 cm2

10. Find the surface area to the nearest tenth. F 546.6 units2 H 1017.9 units2

G 989.6 units2 J 1046.2 units2

11. The radius of a cone is 17 inches long and the slant height is 20 inches. Find the surface area to the nearest tenth.

A 18,158.4 in2 B 1976.1 in2 C 1068.1 in2 D 340 in2

E

AB

D

C

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12 Chapter 12 Test, Form 1 (continued)

12. The area of the base of a prism is 96 square centimeters and the height is 9 centimeters. Find the volume of the prism.

F 288 cm3 G 864 cm3 H 932 cm3 J 7776 cm3

13. The volume of a cylinder is 62.8 cubic meters and the radius is 2 meters. Find the height of the cylinder. Round to the nearest meter.

A 20 m B 10 m C 8 m D 5 m

14. A pyramid has a height of 10 inches and a base with an area of 21 square inches. Find the volume of the pyramid.

F 210 in3 G 105 in3 H 70 in3 J 35 in3

15. Find the volume of the oblique cone. Round to the nearest tenth.

A 1206.4 in3 C 301.6 in3

B 402.1 in3 D 100.5 in3

16. The diameter of a sphere is 42 centimeters. Find the surface area to the nearest tenth.

F 5541.8 cm2 G 2770.9 cm2 H 2167.1 cm2 J 527.8 cm2

17. A sphere has a volume that is 36π cubic meters. Find the radius of the sphere.

A 2 m B 3 m C 6 m D 12 m

18. Which of the following postulates or properties of spherical geometry are true?

F A line has an infinite number of lines parallel to it. G No two lines are parallel. H Alternate interior angles formed by two parallel lines and a

transversal are equal in measure. J Four noncollinear points form two parallel lines.

19. Which of the following describes the two spheres? A congruent C both congruent and similar B similar D neither congruent nor similar

20. The ratio of the side lengths of two cubes is 3:7. Find the ratio of their volumes.

F 3:7 G 9:21 H 9:49 J 27:343

Bonus Find the amount of glass needed to cover the sides of the greenhouse shown. The bottom, front, and back are not glass.

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12


1. What do the dark segments represent in an orthographic drawing? A changes in color C designs on the surface B where paper should be folded D different heights in the surface

For Questions 2 and 3, refer to the figure. 2. Identify the figure. F pyramid H cone G prism J cylinder

3. Identify the shape of a horizontal cross section of the figure. A triangle B ellipse C rectangle D circle

4. The lateral area of a cube is 36 square inches. How long is each edge? F √ � 6 in. G 3 in. H 6 in. J 9 in.

5. Find the surface area of the outside of the open box. A 1920 in2 C 752 in2

B 998 in2 D 400 in2

For Questions 6 and 7, use a right cylinder with a radius of 3 inches and a height of 17 inches. Round to the nearest tenth. 6. Find the lateral area. F 320.4 in2 G 348.7 in2 H 377.0 in2 J 537.2 in2

7. Find the surface area. A 320.4 in2 B 348.7 in2 C 377.0 in2 D 537.2 in2

For Questions 8 and 9, refer to the figure. 8. Find the lateral area. F 144 cm2 H 196 cm2

G 144 + 24 √ � 3 cm2 J 288 cm2

9. Find the surface area. A 144 cm2 B 144 + 24 √ � 3 cm2 C 196 cm2 D 288 cm2

For Questions 10 and 11, refer to the figure. Round to the nearest tenth.

10. Find the lateral area. F 44.0 in2 G 75.4 in2 H 88.0 in2 J 100.5 in2

11. Find the surface area. A 44.0 in2 B 75.4 in2 C 88.0 in2 D 100.5 in2

Chapter 12 Test, Form 2A

Y

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4 cm

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12

12. The surface area of a cube is 96 square feet. Find the volume of the cube.

F 4 ft3 G 16 ft3 H 64 ft3 J 256 ft3

13. A cylinder whose height is 5 meters has a volume of 320π cubic meters. Find the radius of the cylinder.

A 8 m B 12.8 m C 64 m D 201 m

14. A square pyramid has a height that is 8 centimeters long and a base with sides that are each 9 centimeters long. Find the volume of the pyramid.

F 648 cm3 G 324 cm3 H 216 cm3 J 162 cm3

15. Find the volume to the nearest tenth. A 3619.1 m3 C 14,476.5 m3

B 4825.5 m3 D 43,429.4 m3

16. Find the surface area to the nearest tenth. F 4536.5 m2 H 477.5 m2

G 2268.2 m2 J 238.8 m2

17. A sphere has a volume of 972π cubic inches. Find the radius of the sphere.

A 2 in. B 3 in. C 6 in. D 9 in.

18. The shortest distance between any two points in spherical geometry is

F a straight line. G any circle. H a great circle. J a line through the sphere.

19. Two square pyramids are similar. The sides of the bases are 4 inches and 12 inches. The height of the smaller pyramid is 6 inches. Find the height of the larger pyramid.

A 24 in. B 18 in. C 16 in. D 14 in.

20. The ratio of the radii of two similar cylinders is 3:5. The volume of the smaller cylinder is 54π cubic centimeters. Find the volume of the larger cylinder.

F 90π cm3 G 150π cm3 H 250π cm3 J 540π cm3

Bonus Find the surface area of the figure to the nearest tenth.

Chapter 12 Test, Form 2A (continued)

4 ft

12 ft

1 ft

60°

24 m

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12


1. Given the corner view of a figure, which is the top view? A B C D

For Questions 2 and 3, refer to the figure. 2. Identify the figure. F pyramid H cone G prism J cylinder

3. Identify the shape of a vertical cross section of the figure. A rectangle B circle C triangle D parabola

4. Find the lateral area of an equilateral triangular prism if the area of each lateral face is 10 square centimeters.

F 10 √ � 3 cm2 G 30 cm2 H 50 cm2 J 100 cm2

5. The surface area of a rectangular prism is 190 square inches, the length is 10 inches, and the width 3 inches. Find the height.

A 30 in. B 20 in. C 10 in. D 5 in.

For Questions 6–9, use a right cylinder with a radius of 5 centimeters and a height of 22 centimeters. Round to the nearest tenth. 6. Find the lateral area. F 848.2 cm2 G 769.7 cm2 H 691.2 cm2 J 345.6 cm2

7. Find the surface area. A 848.2 cm2 B 769.7 cm2 C 691.2 cm2 D 345.6 cm2

8. Find the volume. F 345.6 cm3 G 691.2 cm3 H 1727.9 cm3 J 2290.2 cm3

For Questions 9–11, refer to the figure. Round to the nearest tenth. 9. Find the lateral area. A 75.4 cm2 C 131.9 cm2

B 103.7 cm2 D 150.8 cm2

10. Find the surface area. G 75.4 cm2 G 103.7 cm2 H 131.9 cm2 J 150.8 cm2

11. Find the volume. A 50.3 cm3 B 69.9 cm3 C 209.7 cm3 J 226.2 cm3

Chapter 12 Test, Form 2B

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12 Chapter 12 Test, Form 2B (continued)

12. The lateral area of a cube is 324 square centimeters. Find the volume of the cube.

F 9 cm3 G 81 cm3 H 729 cm3 J 972 cm3

13. Find the volume of the solid. Round to the nearest tenth.A 31.4 in3 C 125.7 in3

B 41.9 in3 D 502.7 in3

14. A right triangular pyramid has a 12-meter height and a base with legs that are 3 meters and 4 meters long. Find the volume of the triangular pyramid.

F 144 m3 G 72 m3 H 48 m3 J 24 m3

15. Find the volume of the cone. Round to the nearest tenth.A 41,224.0 m3 C 10,306.0 m3

B 20,612.0 m3 D 763.4 m3

16. The surface area of a sphere is 64π square centimeters. Find the radius. F 16 cm G 8 cm H 4 cm J 2 cm

17. A sphere has a 48-centimeter diameter. Find the volume of the sphere. Round to the nearest tenth.

A 463,246.7 cm3 B 57,905.8 cm3 C 28,952.9 cm3 D 7238.2 cm3

18. In spherical geometry, two lines must meet at least how many times?F zero times H two times G one time J an infinite number of times

19. Find the scale factor between the two similar cones.

A 3:8 C 1:2 B 1:3 D 1:4

20. The ratio of the heights of two similar solids is 6:11. Find the ratio of their surface areas.

F 6:11 G 36:121 H 216:1331 J 24:44

Bonus Find the surface area of the frustum of a square pyramid.

10 in.

4 in.

45°

27 m

6 ft

16 ft8 ft

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12 Chapter 12 Test, Form 2C

1. Given the corner view of a figure, sketch the front view.

2. Name the faces of the solid.

3. Sketch the shape of a horizonal cross section of the solid.

4. Find the lateral area of a triangular prism with a height of 8 centimeters, and with bases having sides that measure 4 centimeters, 5 centimeters, and 6 centimeters.

5. Find the surface area of the solid.

6. Find the lateral area of a right cylinder with a diameter of 8.6 yards and a height of 19.4 yards. Round to the nearest tenth.

7. The surface area of a cylinder is 180π square inches and the height is 9 inches. Find the radius.

For Questions 8 and 9, use a regular hexagonal pyramid with base edges of 10 inches and a slant height of 9 inches.



For Questions 10 and 11, use a right circular cone with a radius of 4 feet and a height of 3 feet. Round to the nearest tenth.



1 in.

1 in.

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12

12. The volume of a rectangular prism is 120 cubic feet and the area of the base is 60 square feet. Find the length of a lateral edge of the prism.

13. A cylinder has a 12-foot radius and a 17-foot height. Find the volume of the cylinder. Round to the nearest tenth.

14. A regular hexagonal pyramid has a height that is 15 feet and a base 6 feet on each side. Find the volume of the pyramid. Round to the nearest tenth.


16. Find the surface area of this hemisphere to the nearest tenth.

17. A sphere has a diameter of 7.36 inches long. Find the volume of the sphere. Round to the nearest tenth.

18. What best describes a line in spherical geometry?

19. Determine whether these two cylinders are congruent, similar, or neither.

20. The ratio of the heights of two similar prisms is 2:7. The surface area of the smaller prism is 50 square meters. Find the surface area of the larger prism.

Bonus The length of each side of a cube is 6 inches long. Find the surface area of a sphere inscribed in the cube. Round to the nearest tenth.

12.

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B:

Chapter 12 Test, Form 2C (continued)

11 in.

30°

8 ft 5 ft

4 in.

10 in. 9 in.

3 in.


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12

1. Given the corner view of a figure, sketch the back view.

2. Name the edges of the solid.

3. Sketch the shape of a horizonal cross section of the solid.

4. Find the lateral area of a regular pentagonal prism if the perimeter of the base is 50 inches and the height is 15 inches.

5. Find the surface area of the prism.

6. A right cylinder has a diameter of 23.6 meters and a height of 11.4 meters. Find the lateral area of the cylinder. Round to the nearest tenth.

7. The surface area of a right cylinder is 252π square feet and the height is 11 feet. Find the radius of the cylinder.

For Questions 8 and 9, use a regular octagonal pyramid with base edges 9 feet long, slant height 15 feet, and a base with an apothem of 10.86 feet. 8. Find the lateral area.

9. Find the surface area to the nearest tenth.

For Questions 10 and 11, use a cone with a radius of 5 centimeters and a height of 12 centimeters. Round to the nearest tenth. 10. Find the lateral area.


Chapter 12 Test, Form 2D

9 ft5 ft

6 ft

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12

12. An aquarium is 18 inches long, 8 inches wide, and 14 inches high. The water in it is 4 inches deep. Find the volume of the water.

13. Find the volume of the cylinder. Round to the nearest tenth.

14. A square pyramid has a height that is 51 inches and a base with sides that are each 11 inches long. Find the volume of the pyramid.


16. Find the surface area of the hemisphere. Round to the nearest tenth.

17. A sphere has a radius that is 2.94 centimeters long. Find the volume of the sphere. Round to the nearest tenth.

18. Why do parallel lines not exist in spherical geometry?

19. Determine whether these cubes are congruent, similar, or neither.

20. The ratio of the heights of two similar pyramids is 2:5 and the volume of the smaller pyramid is 100 cubic feet. Find the volume of the larger pyramid.

Bonus The length of each side of a cube is 8 inches long. Find the surface area of a sphere inscribed in the cube. Round to the nearest tenth.

13 in.

Chapter 12 Test, Form 2D (continued)

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B:

25 cm7 cm

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066-076_GEOCRMC12_890521.indd 70066-076_GEOCRMC12_890521.indd 70 6/1/09 10:29:40 AM6/1/09 10:29:40 AM

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12

1. Draw the back view of a figure given its orthographic drawing.

2. Describe the cross section.

3. Find the surface area of the solid. Show the exact solution.

4. Find the lateral area of a triangular prism with a right triangular base with legs that measure 2 feet and 3 feet and a height of 7 feet. Show the exact solution.

5. Find the surface area of the prism.

For Questions 6 and 7, use a right cylinder with a diameter of 96.4 feet and a height of 58.9 feet. Round to the nearest tenth.



For Questions 8 and 9, refer to the solid. Round to the nearest tenth if necessary. 8. Find the lateral area.

9. Find the surface area to the nearest tenth.

For Questions 10 and 11, use a right circular cone with a radius of 7 inches and a height of 8 inches. Round to the nearest tenth. 10. Find the lateral area.


Chapter 12 Test, Form 3

top view left view front view right view 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

13 in.

12 in.

84

3

21√

3

6

10

20


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12

12. Find the volume of the solid.

13. The volume of a cylinder is 96π cubic meters and the height is 6 meters. Find the length of the diameter of this cylinder.

14. Sam is filling a rectangular pan with liquid from a cylindrical can. The can is three-fourths full of water. Determine whether all of the water will fit in the pan. Explain.

15. Find the volume of the solid.

16. Write a formula for the surface area of a hemisphere in terms of π and the radius r.

17. A cone is 9 centimeters deep and 4 centimeters across the top. A single scoop of ice cream, 4 centimeters in diameter, is placed on top of the cone. If the ice cream melts into the cone, determine whether the melted ice cream will fit in the cone. Explain.

18. Is the angle sum of a triangle in spherical geometry always 180°? If not give an example.

19. Determine whether these two pyramids are congruent, similar, or neither.

20. The ratio of the volumes of two similar solids is 1:2. Find the ratio of their surface areas.

Bonus Find the surface area of the solid to the nearest square foot. Do not include the area of the base.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

8 ft

10 ft

6 ft

Chapter 12 Test, Form 3 (continued)

3 cm2 cm

8 cm

10 cm

12 cm

8 in.6 in.

7 in.3 in.2 in.

2 in. 8 in.

13 in.

5 m

5 m

8 m

4 m

5 m8 m

6.4 m8 m


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12 Chapter 12 Extended-Response Test

Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. Explain the difference between the lateral area and the surface area of a prism.

2. Draw an oblique cylinder and a right cylinder.

3. Write a practical application problem involving the surface area or lateral area of a solid figure studied in this chapter.

4. Give the dimensions of two cylinders in which the first has a greater volume than the second, but the second has greater surface area than the first.

5. Draw and label the dimensions of a prism and a pyramid that have the same volume.

6. Write a formula for the volume of this solid in terms of the radius r. Explain.

45°

2r

r

r


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12

x

yxO

B

A F G

C

E

D

97°

49°

8.9

r

Q R

P

40°

40°

4.2

12.6

9.5

a

A

B E

C

F 1. Which method could you use to prove

−−−

BE � −−

AC if AF = BF? (Lesson 4-5)

A Show that �ABE � �BAC by SSS, then

−−−

BE � −−

AC by CPCTC. B Show that �ABE � �BAC by ASA, then

−−−

BE � −−

AC by CPCTC. C Show that �BFE � �AFC by SAS, then

−−−

BE � −−

AC by CPCTC. D Show that �ABE � �BAC by AAS, then

−−−

BE � −−

AC by CPCTC.

2. Find a. (Lesson 7-3)

F 28.5 H 12.6G 6.3 J 14

3. Find r. (Lesson 8-6) A about 34.0 C about 11.8B about 8.9 D about 6.6

4. A square has side length 18 centimeters. Find the area of the square. (Lesson 11-1)

F 36 cm2 G 40 cm2 H 81 cm2 J 324 cm2

5. What can you assume from the figure? (Lesson 10-3)

A �ABC is isosceles.B �ABC is equilateral.C DF = EGD radius of �O = x + y

6. Points D, E, and F are on a circle so that m � DEF = 210. If K is the center of the circle, what is m ∠DKF? (Lesson 10-2)

F 210 G 105 H 70 J 35

7. Which net could be folded into a triangular prism? (Lesson 12-1)

A B C D

8. Find the surface area of a square pyramid with a height of 9 centimeters and base with a side measuring 24 centimeters. (Lesson 12-3)

F 1296 cm2 G 1806 cm2 H 2016 cm2 J 8640 cm2

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

8. F G H J

A B C D

Standardized Test Practice(Chapters 1–12)

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

A B C D


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12

13.

14.

9. Find y to the nearest centimeter. (Lesson 8-6)

A 19 cm C 34 cmB 28 cm D 37 cm

10. A plane figure is the locus of all points in a plane equidistant from point B. What is the shape of this figure? (Lesson 10-1)

F square G cylinder H rhombus J circle

11. Find m∠C. (Lesson 10-6)

A 18º C 28ºB 25º D 60º

12. Find the area of the figure. (Lesson 11-1)

F 76 cm2 H 88 cm2

G 80 cm2 J 92 cm2

73°

32 cm

25 cmy

Z Y

X

9.

10.

11.

12. F G H J

A B C D

F G H J

A B C D

13. Quadrilateral PQSR is a rectangle. Find a. (Lesson 6-4)

14. Find x. Assume that segments that appear tangent are tangent.(Lesson 10-5)

2x + 8

5x - 7

M

N

(2a - 1)°(6a - 21)°

P Q

SR

Standardized Test Practice (continued)

6 cm

4 cm

4 cm

7 cm

3 cm2 cm

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the answer in a column box

and then shading in the appropriate circle that corresponds to that entry.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

88° 32°

E

A

D

BC


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12

5√3 cm

60° 60°

60°

15 cm

15

3018

x

50°

3 mT

S

R

15. Find the length of � SR to the nearest tenth. (Lesson 10-2)

16. Find x. (Lesson 10-7)

17. Find the area of the shaded region to the nearest tenth. (Lesson 11-3)

18. Identify the solid. (Lesson 12-1)

19. A right circular cone has a slant height of 15 inches and a radius that is 25 inches long. Find the surface area of the cone. Round to the nearest tenth. (Lesson 12-3)

20. A ball has a diameter of 26.5 centimeters. Find the surface area of the ball. Round to the nearest tenth. (Lesson 12-6)

21. Find the following measurements for a sphere with a diameter of 66 meters. Round to the nearest tenth. (Lesson 12-6)

a. surface area

b. circumference of the great circle

c. area of the great circle

d. surface area of the hemisphere

15.

16.

17.

18.

19.

20.

21a.

b.

c.

d.

Standardized Test Practice (continued)

Part 3: Short Response

Instructions: Write your answer in the space provided.

066-076_GEOCRMC12_890521.indd 76066-076_GEOCRMC12_890521.indd 76 6/1/09 11:58:42 AM6/1/09 11:58:42 AM

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Chapter 12 A1 Glencoe Geometry

Chapter Resources

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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Answers (Anticipation Guide and Lesson 12-1)

A01-A26_GEOCRMC12_890521.indd A1A01-A26_GEOCRMC12_890521.indd A1 6/14/08 9:20:38 PM6/14/08 9:20:38 PM

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Answers (Lesson 12-1)


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o sk

etch

eac

h p

rism

.

1. r

ecta

ngu

lar

pris

m 3

un

its

hig

h,

2. t

rian

gula

r pr

ism

3 u

nit

s h

igh

, wh

ose

base

s 3

un

its

lon

g, a

nd

2 u

nit

s w

ide

are

rig

ht

tria

ngl

es w

ith

leg

s 2

un

its

and

4

un

its

lon

g

Use

iso

met

ric

dot

pap

er a

nd

eac

h o

rth

ogra

ph

ic d

raw

ing

to s

ket

ch a

sol

id.

3.

top

view

left

view

front

vie

wrig

ht v

iew

4.

top

view

left

view

front

vie

wrig

ht v

iew

Sk

etch

th

e cr

oss

sect

ion

fro

m a

ver

tica

l sl

ice

of e

ach

fig

ure

.

5.

6.

7. S

PHER

ES C

onsi

der

the

sph

ere

in E

xerc

ise

5. B

ased

on

th

e cr

oss

sect

ion

res

ult

ing

from

a

hor

izon

tal

and

a ve

rtic

al s

lice

of

the

sph

ere,

mak

e a

con

ject

ure

abo

ut

all

sph

eric

al

cros

s se

ctio

ns.

8. M

INER

ALS

Pyr

ite,

als

o kn

own

as

fool

’s g

old,

can

for

m c

ryst

als

that

are

per

fect

cu

bes.

S

upp

ose

a ge

mol

ogis

t w

ants

to

cut

a cu

be o

f py

rite

to

get

a sq

uar

e an

d a

rect

angl

ar f

ace.

W

hat

cu

ts s

hou

ld b

e m

ade

to g

et e

ach

of

the

shap

es?

Illu

stra

te y

our

answ

ers.

Prac

tice

Rep

resen

tati

on

s o

f Th

ree-D

imen

sio

nal

Fig

ure

s

12-1

cir

cle

trap

ezo

id

All s

ph

eri

cal

cro

ss s

ecti

on

s a

re c

ircle

s.

a c

ut

para

llel

to

the b

ases t

o g

et

a s

qu

are

a c

ut

thro

ug

h

dia

go

nally o

pp

osit

e

top

an

d b

ott

om

ed

ges

to g

et

a r

ecta

ng

le

001_

025_

GE

OC

RM

C12

_890

521.

indd

84/

10/0

89:

13:2

7P

M


NA

ME

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PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-1

Cha

pte

r 12

9

G

lenc

oe G

eom

etry

1. L

AB

ELS

Jam

al r

emov

es t

he

labe

l fr

om a

cy

lin

dric

al s

oup

can

to

earn

poi

nts

for

h

is s

choo

l. S

ketc

h t

he

shap

e of

th

e la

bel.

2. B

LOC

KS

Mar

got’s

th

ree-

year

-old

son

m

ade

the

mag

net

ic b

lock

scu

lptu

re

show

n b

elow

in

cor

ner

vie

w.

D

raw

th

e ri

ght

view

of

the

scu

lptu

re.

3. C

UB

ES N

ath

an m

arks

th

e m

idpo

ints

of

thre

e ed

ges

of a

cu

be a

s sh

own

.H

e th

en s

lice

s th

e cu

be a

lon

g a

plan

e th

at c

onta

ins

thes

e th

ree

poin

ts.

Des

crib

e th

e re

sult

ing

cros

s se

ctio

n.

4. E

NG

INEE

RIN

G S

teph

anie

nee

ds a

n

obje

ct w

hos

e to

p vi

ew i

s a

circ

le a

nd

wh

ose

left

an

d fr

ont

view

s ar

e sq

uar

es.

Des

crib

e an

obj

ect

that

wil

l sa

tisf

y th

ese

con

diti

ons.

5. D

ESK

SU

PPO

RTS

Th

e fi

gure

sh

ows

the

supp

ort

for

a de

sk.

a. D

raw

th

e to

p vi

ew.

b.

Dra

w t

he

fron

t vi

ew.

c. D

raw

th

e ri

ght

view

.

Wor

d Pr

oble

m P

ract

ice

Rep

resen

tati

on

s o

f Th

ree-D

imen

sio

nal

Fig

ure

s

12-1 reg

ula

r h

exag

on

Sam

ple

an

sw

er:

A c

ylin

der

wit

h

its h

eig

ht

eq

ual

to i

ts d

iam

ete

r.

001_

025_

GE

OC

RM

C12

_890

521.

indd

94/

10/0

89:

13:3

3P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

1

0

Gle

ncoe

Geo

met

ry

Dra

win

g S

olid

s o

n I

so

metr

ic D

ot

Pap

er

Isom

etri

c do

t pa

per

is h

elpf

ul

for

draw

ing

soli

ds. R

emem

ber

to u

se

dash

ed l

ines

for

hid

den

edg

es.

For

eac

h s

olid

sh

own

, dra

w a

not

her

sol

id w

hos

e d

imen

sion

s ar

e tw

ice

as l

arge

.

1.

2.

3.

4.

5.

6.

Enri

chm

ent

12-1

001_

025_

GE

OC

RM

C12

_890

521.

indd

104/

10/0

89:

13:3

9P

M


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-1

Cha

pte

r 12

1

1

Gle

ncoe

Geo

met

ry

Th

e sc

ien

ce o

f pe

rspe

ctiv

e dr

awin

g st

udi

es h

ow t

o dr

aw a

th

ree-

dim

ensi

onal

obj

ect

on a

tw

o-di

men

sion

al p

age.

Th

is s

cien

ce b

ecam

e h

igh

ly

refi

ned

du

rin

g th

e R

enai

ssan

ce w

ith

th

e w

ork

of a

rtis

ts s

uch

as

Alb

rech

t D

üre

r an

d L

eon

ardo

da

Vin

ci.

Tod

ay, c

ompu

ters

are

oft

en u

sed

to m

ake

pers

pect

ive

draw

ings

, par

ticu

larl

y el

abor

ate

grap

hic

s u

sed

in t

elev

isio

n a

nd

mov

ies.

Th

e th

ree-

dim

ensi

onal

co

ordi

nat

es o

f ob

ject

s ar

e fi

gure

d. T

hen

alg

ebra

is

use

d to

tra

nsf

orm

th

ese

into

tw

o-di

men

sion

al c

oord

inat

es. T

he

grap

h o

f th

ese

new

coo

rdin

ates

is

call

ed a

pro

ject

ion

.

Th

e fo

rmu

las

belo

w w

ill

draw

on

e ty

pe o

f pr

ojec

tion

in

wh

ich

th

e y-

axis

is

draw

n h

oriz

onta

lly,

th

e z-

axis

ver

tica

lly,

an

d th

e x-

axis

at

an a

ngl

e of

a˚

wit

h t

he

y-ax

is. I

f th

e th

ree-

dim

ensi

onal

coo

rdin

ates

of

a po

int

are

(x, y

, z),

th

en t

he

proj

ecti

on c

oord

inat

es (

X, Y

) ar

e gi

ven

by

X =

x(-

cos

a) +

y a

nd

Y =

x(-

sin

a)

+ z

.A

lth

ough

th

is t

ype

of p

roje

ctio

n g

ives

a f

airl

y go

od p

ersp

ecti

ve d

raw

ing,

it

does

dis

tort

som

e le

ngt

hs.

1. T

he

draw

ing

wit

h t

he

coor

din

ates

giv

en b

elow

is

a cu

be.

A

(5, 0

, 5),

B(5

, 5, 5

), C

(5, 5

, 0),

D(5

, 0, 0

),

E(0

, 0, 5

), F

(0, 5

, 5),

G(0

, 5, 0

), H

(0, 0

, 0)

U

se t

he

form

ula

s ab

ove

to f

ind

the

proj

ecti

on c

oord

inat

es o

f ea

ch

poin

t, u

sin

g a

= 4

5. R

oun

d pr

ojec

tion

coo

rdin

ates

to

the

nea

rest

in

tege

r. G

raph

th

e cu

be o

n a

gra

phin

g ca

lcu

lato

r. M

ake

a sk

etch

of

th

e di

spla

y.

A'(_

_, _

_) B

'(__,

__)

C′(_

_, _

_) D

′(__,

__)

E' (_

_, _

_) F

'(__,

__)

G′(_

_, _

_) H

′(__,

__)

2. T

he

poin

ts A

(10,

2, 0

), B

(10,

10,

0),

C(2

, 10,

0),

an

d D

(3, 3

, 4)

are

vert

ices

of

a py

ram

id.

Fin

d th

e pr

ojec

tion

coo

rdin

ates

, usi

ng

a =

25.

Rou

nd

coor

din

ates

to

the

nea

rest

in

tege

r.

Th

en g

raph

th

e py

ram

id o

n a

gra

phin

g ca

lcu

lato

r by

dra

win

g −

−−

A

′B′ ,

−−

−

B

′C′ ,

−−

−

C

′D′ ,

−−

−

D

′A′ ,

and

−−

−

D

′B′ .

Mak

e a

sket

ch o

f th

e di

spla

y.

A′(_

_, _

_) B

′(__,

__)

C′(_

_, _

_) D

′(__,

__)

A'(-

7,

-4),

B'(1

, -

4),

C'(8

, -

1),

D'(0

, 3);

fo

r sketc

hes,

see s

tud

en

ts’

wo

rk.

Gra

phin

g Ca

lcul

ator

Act

ivit

yP

ers

pecti

ve D

raw

ing

s

A'(-

4,

1),

B'(1

, 1),

C'(1

, -

4),

D'(-

4,

-4),

E'(0

, 5),

F'(5

, 5),

G'(5

, 0)

H'(0

,0);

F

or

sketc

hes,

see s

tud

en

ts’

wo

rk.

12-1

EF

AB

D

H

C

G

001_

025_

GE

OC

RM

C12

_890

521.

indd

116/

14/0

82:

45:5

7P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

1

2

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

Su

rface A

reas o

f P

rism

s a

nd

Cylin

ders

12-2

Late

ral a

nd

Su

rfac

e A

reas

of

Pris

ms

In a

sol

id f

igu

re,

face

s th

at a

re n

ot b

ases

are

lat

eral

fac

es. T

he

late

ral

area

is

the

sum

of

the

area

of

the

late

ral

face

s. T

he

surf

ace

area

is

the

sum

of

the

late

ral

area

an

d th

e ar

ea o

f th

e ba

ses.

La

tera

l A

rea

of

a P

ris

m

If a

prism

ha

s a

la

tera

l a

rea

of

L sq

ua

re u

nits,

a h

eig

ht

of

h u

nits,

an

d e

ach

ba

se

ha

s a

pe

rim

ete

r o

f P

un

its,

the

n L

= P

h.

Su

rfa

ce

Are

a

of

a P

ris

m

If a

prism

ha

s a

su

rfa

ce

are

a o

f S

sq

ua

re u

nits,

a la

tera

l a

rea

of

L sq

ua

re u

nits,

an

d e

ach

ba

se

ha

s a

n a

rea

of

B s

qu

are

un

its,

the

n

S =

L +

2B

or

S =

Ph

+ 2

B

F

ind

th

e la

tera

l an

d s

urf

ace

area

of

the

regu

lar

pen

tago

nal

p

rism

ab

ove

if e

ach

bas

e h

as a

per

imet

er o

f 75

cen

tim

eter

s an

d t

he

hei

ght

is

10 c

enti

met

ers.

L =

Ph

Late

ral are

a o

f a p

rism

=

75(

10)

P =

75,

h =

10

=

750

M

ultip

ly.

Th

e la

tera

l ar

ea i

s 75

0 sq

uar

e ce

nti

met

ers

and

the

surf

ace

area

is

abou

t 15

24.2

squ

are

cen

tim

eter

s.

Exer

cise

sF

ind

th

e la

tera

l ar

ea a

nd

su

rfac

e ar

ea o

f ea

ch p

rism

. Rou

nd

to

the

nea

rest

ten

th

if n

eces

sary

.

1.

4 m

3 m

10 m

2.

15 in

.

10 in

.

8 in

.

3.

6 in

.18

in.

4.

20 c

m

10 c

m10

cm

12 c

m8

cm9

cm

5.

4 in

.

4 in

.

12 in

.

6.

4 m

16 m

Exam

ple

pent

agon

al p

rismal

titud

e

late

ral

edge

late

ral

face

S =

L +

2B

=

750

+ 2

( 1 −

2 a

P)

=

750

+ (

7.5

−

tan

36°

) (75)

≈

152

4.2

tan

36°

= 7.

5 −

a

a

=

7.5

−

tan

36°

a 15 c

m36°

L =

120 m

2; S

= 1

32 m

2

L =

540 i

n2 (

10 i

n.

× 8 i

n.

base);

S

= 7

00 i

n2

L =

460 i

n2

(8 i

n.

× 1

5 i

n.

base)

or

L =

400 i

n2

(10 in

. ×

15 in

. b

ase)

or

L =

540 i

n2; S

≈ 6

63.9

in

2

L =

588 c

m2; S

= 8

28 c

m2

L =

128 in

2 (

recta

ng

ula

r b

ase)

or

L =

384 m

2; S

= 4

67.1

m2

L =

192 i

n2 (

sq

uare

base);

S =

224 i

n2

001_

025_

GE

OC

RM

C12

_890

521.

indd

126/

2/09

4:25

:03

PM


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-2

Cha

pte

r 12

13

Gle

ncoe

Geo

met

ry

Late

ral a

nd

Su

rfac

e A

reas

of

Cyl

ind

ers

A c

ylin

der

is

a so

lid

wit

h b

ases

th

at a

re c

ongr

uen

t ci

rcle

s ly

ing

in p

aral

lel

plan

es.

Th

e ax

is o

f a

cyli

nde

r is

th

e se

gmen

t w

ith

en

dpoi

nts

at

the

cen

ters

of

th

ese

circ

les.

For

a r

igh

t cy

lin

der

, th

e ax

is i

s al

so t

he

alti

tude

of

th

e cy

lin

der.

La

tera

l A

rea

of

a C

yli

nd

er

If a

cylin

de

r h

as a

la

tera

l a

rea

of

L sq

ua

re u

nits,

a h

eig

ht

of

h u

nits,

an

d a

ba

se

ha

s a

ra

diu

s o

f r

un

its,

the

n L

= 2

πrh

.

Su

rfa

ce

Are

a

of

a C

yli

nd

er

If a

cylin

de

r h

as a

su

rfa

ce

are

a o

f S

sq

ua

re u

nits,

a h

eig

ht

of

h u

nits,

an

d a

ba

se

ha

s a

ra

diu

s o

f r

un

its,

the

n S

= L

+ 2

B o

r 2

πrh

+ 2

πr2

.

F

ind

th

e la

tera

l an

d s

urf

ace

area

of

the

cyli

nd

er. R

oun

d t

o th

e n

eare

st t

enth

.If

d =

12

cm, t

hen

r =

6 c

m.

L =

2πrh

Late

ral are

a o

f a c

ylin

der

=

2π(6

)(14

) r

= 6

, h

= 1

4

≈

527

.8

Use a

calc

ula

tor.

S =

2πrh

+ 2

πr2

Surf

ace a

rea o

f a c

ylin

der

≈

527

.8 +

2π(6

)2 2

πrh

≈ 5

27.8

, r

= 6

≈

754

.0

Use a

calc

ula

tor.

Th

e la

tera

l ar

ea i

s ab

out

527.

8 sq

uar

e ce

nti

met

ers

and

the

surf

ace

area

is

abou

t 75

4.0

squ

are

cen

tim

eter

s.

Exer

cise

sF

ind

th

e la

tera

l ar

ea a

nd

su

rfac

e ar

ea o

f ea

ch c

ylin

der

. Rou

nd

to

the

nea

rest

te

nth

. 1

.

12 c

m

4 cm

2.

6 in

.10

in.

3.

6 cm

3 cm

3 cm

4.

20 c

m

8 cm

5.

12 m

4 m

6.

2

m

1 m

radi

us o

f bas

eax

is

base

base

heig

ht

Exam

ple

14 c

m

12 c

m

12-2

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Su

rface A

reas o

f P

rism

s a

nd

Cylin

ders

L ≈

301.6

cm

2; S

≈ 4

02.1

cm

2L ≈

377.0

in

2; S

≈ 6

03.2

in

2

L ≈

113.1

cm

2; S

≈ 1

69.6

cm

2L ≈

502.7

cm

2; S

≈ 6

03.2

cm

2

L ≈

150.8

m2; S

≈ 3

77.0

m2

L ≈

12.6

m2; S

≈ 3

7.7

m2

001_

025_

GE

OC

RM

C12

_890

521.

indd

135/

30/0

93:

07:0

0P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

1

4

Gle

ncoe

Geo

met

ry

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

pri

sm. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

2.

3.

4.

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

cyl

ind

er. R

oun

d t

o th

e n

eare

st

ten

th.

5.

6.

7.

8.

12 y

d

12 y

d

10 y

d8

m

6 m

12 m

10 in

.5

in.

6 in

.

8 in

.

9 cm

9 cm7.8

cm9

cm

12 c

m

L

= 480 y

d2 (

sq

uare

base)

L =

240 m

2 (

8 ×

12 b

ase)

L =

528 y

d2 (

recta

ng

ula

r b

ase)

L =

288 m

2 (

12 ×

6 b

ase)

S =

768 y

d2

L =

336 m

2 (

8 ×

6 b

ase)

S =

432 m

2

L

= 120 i

n2

L =

324 c

m2

S

= 168 i

n2

S =

394.2

cm

2

Skill

s Pr

acti

ceS

urf

ace A

reas o

f P

rism

s a

nd

Cylin

ders

12-2

L

≈ 377.0

in

2

L ≈

25.1

m2

S

≈ 603.2

in

2

S ≈

50.3

m2

L

≈ 37.7

yd

2

L ≈

603.2

in

2

S

≈ 94.2

yd

2

S ≈

1005.3

in

2

8 in

.

12 in

.

2 yd

3 yd12

in.

10 in

.

2 m

2 m

001_

025_

GE

OC

RM

C12

_890

521.

indd

145/

30/0

93:

07:0

4P

M


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-2

Cha

pte

r 12

15

Gle

ncoe

Geo

met

ry

12-2

Prac

tice

Su

rface A

reas o

f P

rism

s a

nd

Cylin

ders

Fin

d t

he

late

ral

and

su

rfac

e ar

ea o

f ea

ch p

rism

. Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

. 1

.

15 c

m

15 c

m

32 c

m

2.

8 ft

10 ft

5 ft

3.

2 m

11 m

4.

4 yd

4 yd

9.5

yd

5 yd

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

cyl

ind

er. R

oun

d t

o th

e n

eare

st

ten

th.

5.

5 ft 7

ft

6.

4 m

8.5

m

7.

19 in

.

17 in

. 8.

12 m

30 m

L =

1920 c

m2 (

sq

uare

base)

or

L =

224.3

ft2

;L =

1410 c

m2 (

recta

ng

ula

r b

ase);

S =

264.3

ft2

S =

2370 c

m2

L =

132 m

2; S

≈ 152.8

m2

L =

123.5

yd

2; S

≈ 1

39.1

yd

2

L ≈

219.9

ft2

;

L ≈

106.8

m2;

S ≈

377.0

ft2

S

≈ 131.9

m2

L ≈

1014.7

in

2;

L ≈

2261.9

m2;

S ≈

1581.8

in

2

S ≈

3166.7

m2

001_

025_

GE

OC

RM

C12

_890

521.

indd

153/

5/13

10:0

8:32

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

1

6

Gle

ncoe

Geo

met

ry

1. L

OG

OS

Th

e Z

com

pan

y sp

ecia

lize

s in

ca

rin

g fo

r ze

bras

. Th

ey w

ant

to m

ake

a 3-

dim

ensi

onal

“Z

” to

pu

t in

fro

nt

of t

hei

r co

mpa

ny

hea

dqu

arte

rs. T

he

“Z”

is

15 i

nch

es t

hic

k an

d th

e pe

rim

eter

of

the

base

is

390

inch

es.

15"

W

hat

is

the

late

ral

surf

ace

area

of

this

“Z

”?

2. S

TAIR

WEL

LS M

anag

emen

t de

cide

s to

en

clos

e st

airs

con

nec

tin

g th

e fi

rst

and

seco

nd

floo

rs o

f a

park

ing

gara

ge i

n a

st

airw

ell

shap

ed l

ike

an o

bliq

ue

rect

angu

lar

pris

m.

16

ft

20

ft

15

ft

9 ft

W

hat

is

the

late

ral

surf

ace

area

of

the

stai

rwel

l?

3. C

AK

ES A

cak

e is

a r

ecta

ngu

lar

pris

m

wit

h h

eigh

t 4

inch

es a

nd

base

12

inch

es

by 1

5 in

ches

. Wal

lace

wan

ts t

o ap

ply

fros

tin

g to

th

e si

des

and

the

top

of t

he

cake

. Wh

at i

s th

e su

rfac

e ar

ea o

f th

e pa

rt o

f th

e ca

ke t

hat

wil

l h

ave

fros

tin

g?

4. E

XH

AU

ST P

IPES

An

exh

aust

pip

e is

sh

aped

lik

e a

cyli

nde

r w

ith

a h

eigh

t of

50

in

ches

an

d a

radi

us

of 2

in

ches

. Wh

at

is t

he

late

ral

surf

ace

area

of

the

exh

aust

pi

pe?

Rou

nd

you

r an

swer

to

the

nea

rest

h

un

dred

th.

5. T

OW

ERS

A c

ircu

lar

tow

er i

s m

ade

by

plac

ing

one

cyli

nde

r on

top

of

anot

her

. B

oth

cyl

inde

rs h

ave

a h

eigh

t of

18

inch

es. T

he

top

cyli

nde

r h

as a

rad

ius

of

18 i

nch

es a

nd

the

bott

om c

ylin

der

has

a

radi

us

of 3

6 in

ches

.

18

in.

18

in.

a. W

hat

is

the

tota

l su

rfac

e ar

ea o

f th

e to

wer

? R

oun

d yo

ur

answ

er t

o th

e n

eare

st h

un

dred

th.

b.

An

oth

er t

ower

is

con

stru

cted

by

plac

ing

the

orig

inal

tow

er o

n t

op o

f an

oth

er c

ylin

der

wit

h a

hei

ght

of

18 i

nch

es a

nd

a ra

diu

s of

54

inch

es.

Wh

at i

s th

e to

tal

surf

ace

area

of

the

new

tow

er?

Rou

nd

you

r an

swer

to

the

nea

rest

hu

ndr

edth

.

12-2

Wor

d Pr

oble

m P

ract

ice

Su

rface A

reas o

f P

rism

s a

nd

Cylin

ders

5850 i

n2

840 f

t2

396 i

n2

628.3

2 i

n2

14,2

50.2

6 i

n2

30,5

36.2

8 i

n2

001_

025_

GE

OC

RM

C12

_890

521.

indd

164/

10/0

89:

14:0

7P

M


NA

ME

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PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-2

Cha

pte

r 12

17

Gle

ncoe

Geo

met

ry

Enri

chm

ent

12-2

Min

imiz

ing

Co

st

in M

an

ufa

ctu

rin

gS

upp

ose

that

a m

anu

fact

ure

r w

ants

to

mak

e a

can

th

at h

as a

vol

um

e of

40

cu

bic

inch

es. T

he

cost

to

mak

e th

e ca

n i

s 3

cen

ts p

er s

quar

e in

ch f

or

the

top

and

bott

om a

nd

1 ce

nt

per

squ

are

inch

for

th

e si

de.

1. W

rite

th

e va

lue

of h

in

ter

ms

of r

, giv

en v

= π

r2h

.

h =

40

−

πr2

2. W

rite

a f

orm

ula

for

th

e co

st i

n t

erm

s of

r.

C

= 3

(2π

r2)

+ 1

(2π

r � 4

0

−

πr2

) o

r 6

πr2

+ 8

0

−

r

3. U

se a

gra

phin

g ca

lcu

lato

r to

gra

ph t

he

form

ula

, let

tin

g Y

1 re

pres

ent

the

cost

an

d X

re

pres

ent

r. U

se t

he

grap

h t

o es

tim

ate

the

poin

t at

wh

ich

th

e co

st i

s m

inim

ized

.

Th

e m

inim

um

heig

ht

is 7

.71 i

nch

es,

wh

ich

giv

es a

min

imu

m c

ost

of

93.

4. R

epea

t th

e pr

oced

ure

usi

ng

2 ce

nts

per

squ

are

inch

for

th

e to

p an

d bo

ttom

an

d 4

cen

ts p

er s

quar

e in

ch f

or t

he

top

and

bott

om.

2

cen

ts:

C =

4π

r2 +

80

−

r =

5.8

8 i

n.

; 4 c

en

ts:

C =

8π

r2 +

80

−

r =

9.3

4 i

n.

5. W

hat

wou

ld y

ou e

xpec

t to

hap

pen

as

the

cost

of

the

top

and

bott

om i

ncr

ease

s?

S

ee s

tud

en

ts’

wo

rk.

Sam

ple

an

sw

er:

Th

e m

an

ufa

ctu

rer

mig

ht

make t

he

can

s t

aller

an

d n

arr

ow

er.

6. C

ompu

te t

he

tabl

e fo

r th

e co

st v

alu

e gi

ven

. Wh

at h

appe

ns

to t

he

hei

ght

of t

he

can

as

the

cost

of

the

top

and

bott

om i

ncr

ease

s?

T

he h

eig

ht

incre

ases a

s t

he c

ost

of

the t

op

an

d b

ott

om

go

up

.

Co

st

To

pC

os

tM

inim

um

& B

ott

om

Cy

lin

de

rh

2 ce

nts

1 ce

nt

5.8

8

3 ce

nts

1 ce

nt

7.7

1

4 ce

nts

1 ce

nt

9.3

4

5 ce

nts

1 ce

nt

10.8

4

6 ce

nts

1 ce

nt

12.2

4

r

h

001_

025_

GE

OC

RM

C12

_890

521.

indd

176/

2/09

4:28

:37

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

1

8

Gle

ncoe

Geo

met

ry

12-3

Stud

y G

uide

and

Inte

rven

tion

Su

rface A

reas o

f P

yra

mid

s a

nd

Co

nes

Late

ral a

nd

Su

rfac

e A

reas

of

Pyra

mid

s A

pyr

amid

is

a so

lid

wit

h a

pol

ygon

bas

e. T

he

late

ral

face

s in

ters

ect

in a

com

mon

po

int

know

n a

s th

e ve

rtex

. Th

e al

titu

de i

s th

e se

gmen

t fr

om t

he

vert

ex t

hat

is

perp

endi

cula

r to

th

e ba

se. F

or a

reg

ula

r p

yram

id,

the

base

is

a re

gula

r po

lygo

n a

nd

the

alti

tude

has

an

en

dpoi

nt

at

the

cen

ter

of t

he

base

. All

th

e la

tera

l ed

ges

are

con

gru

ent

and

all

the

late

ral

face

s ar

e co

ngr

uen

t is

osce

les

tria

ngl

es. T

he

hei

ght

of e

ach

la

tera

l fa

ce i

s ca

lled

th

e sl

ant

hei

ght.

La

tera

l A

rea

of

a R

eg

ula

r P

yra

mid

Th

e la

tera

l a

rea

L o

f a

re

gu

lar

pyra

mid

is L

= 1

−

2 P

ℓ, w

he

re ℓ

is t

he

sla

nt

he

igh

t a

nd

P is t

he

pe

rim

ete

r o

f th

e b

ase

.

Su

rfa

ce

Are

a o

f

a R

eg

ula

r P

yra

mid

Th

e s

urf

ace

are

a S

of

a r

eg

ula

r p

yra

mid

is S

=

1

−

2 P

ℓ +

B,

wh

ere

ℓ is t

he

sla

nt

he

igh

t, P

is t

he

pe

rim

ete

r o

f th

e b

ase

,

an

d B

is t

he

are

a o

f th

e b

ase

.

F

or t

he

regu

lar

squ

are

pyr

amid

ab

ove,

fin

d t

he

late

ral

area

an

d

surf

ace

area

if

the

len

gth

of

a si

de

of t

he

bas

e is

12

cen

tim

eter

s an

d t

he

hei

ght

is

8 ce

nti

met

ers.

Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

Fin

d th

e sl

ant

hei

ght.

ℓ2 =

62

+ 8

2 P

yth

agore

an T

heore

m

ℓ2 =

100

S

implif

y.

ℓ =

10

Take t

he p

ositiv

e s

quare

root

of

each s

ide.

Th

e la

tera

l ar

ea i

s 24

0 sq

uar

e ce

nti

met

ers,

an

d th

e su

rfac

e ar

ea i

s 38

4 sq

uar

e ce

nti

met

ers.

Exer

cise

sF

ind

th

e la

tera

l ar

ea a

nd

su

rfac

e ar

ea o

f ea

ch r

egu

lar

pyr

amid

. Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.

15 c

m

20 c

m

2.

45°

8 ft

3.

60°

10 c

m

4.

6 in

.8.

7 in

.15

in.

late

ral e

dge

base

slan

t hei

ght

heig

ht

Exam

ple

L =

1 −

2 P

ℓ Late

ral are

a o

f a r

egula

r pyra

mid

=

1 −

2 (4

8)(1

0)

P =

4 �

12

or

48, ℓ

= 1

0

=

240

S

implif

y.

S =

1 −

2 P

ℓ +

B

Surf

ace a

rea o

f a r

egula

r pyra

mid

=

240

+ 1

44

1

−

2 P

ℓ =

240, B

= 1

2 ·

12 o

r 144

=

384

L ≈

450 c

m2; S

≈ 5

47.4

cm

2

L ≈

362.0

ft2

; S ≈

618.0

ft2

L ≈

266.7

cm

2; S

≈ 4

00.0

cm

2

L ≈

326.3

in

2; S

≈ 4

56.8

in

2

001_

025_

GE

OC

RM

C12

_890

521.

indd

183/

27/1

010

:44:

01P

M


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-3

Cha

pte

r 12

19

Gle

ncoe

Geo

met

ry

12-3

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Su

rface A

reas o

f P

yra

mid

s a

nd

Co

nes

Late

ral a

nd

Su

rfac

e A

reas

of

Co

nes

A c

one

has

a

circ

ula

r ba

se a

nd

a ve

rtex

. Th

e ax

is o

f th

e co

ne

is t

he

segm

ent

wit

h e

ndp

oin

ts a

t th

e ve

rtex

an

d th

e ce

nte

r of

th

e ba

se. I

f th

e ax

is i

s al

so t

he

alti

tude

, th

en t

he

con

e is

a

righ

t co

ne.

If

the

axis

is

not

th

e al

titu

de, t

hen

th

e co

ne

is a

n o

bli

qu

e co

ne.

La

tera

l A

rea

of

a C

on

e

Th

e la

tera

l a

rea

L o

f a

rig

ht

circu

lar

co

ne

is L

= π

r�,

wh

ere

r is

the

ra

diu

s a

nd

� is t

he

sla

nt

he

igh

t.

Su

rfa

ce

Are

a o

f

a C

on

e

Th

e s

urf

ace

are

a S

of

a r

igh

t co

ne

is S

= π

r� +

πr2

, w

he

re r

is

the

ra

diu

s a

nd

� is t

he

sla

nt

he

igh

t.

F

or t

he

righ

t co

ne

abov

e, f

ind

th

e la

tera

l ar

ea a

nd

su

rfac

e ar

ea i

f th

e ra

diu

s is

6 c

enti

met

ers

and

th

e h

eigh

t is

8 c

enti

met

ers.

Rou

nd

to

the

nea

rest

te

nth

if

nec

essa

ry.

Fin

d th

e sl

ant

hei

ght.

ℓ2 =

62

+ 8

2 P

yth

agore

an T

heore

m

ℓ2 =

100

S

implif

y.

ℓ =

10

Take t

he p

ositiv

e s

quare

root

of

each s

ide.

Th

e la

tera

l ar

ea i

s ab

out

188.

5 sq

uar

e ce

nti

met

ers

and

the

surf

ace

area

is

abou

t 30

1.6

squ

are

cen

tim

eter

s.

Exer

cise

sF

ind

th

e la

tera

l ar

ea a

nd

su

rfac

e ar

ea o

f ea

ch c

one.

Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.

9 cm

12 c

m

2.

5 ft

30°

3.

12 c

m

13 c

m

4.

4 in

.

45°

axis

base

base

�sl

ant h

eigh

t

right

con

eob

lique

con

e

altit

ude

VV

Exam

ple

L =

πrℓ

Late

ral are

a o

f a r

ight

cone

=

π(6

)(10

) r

= 6

, ℓ

= 1

0

≈

188

.5

Sim

plif

y.

S =

πrℓ

+ π

r2 S

urf

ace a

rea o

f a r

ight

cone

≈

188

.5 +

π(6

2 )

πrℓ

≈ 1

88.5

, r

= 6

≈

301

.6

Sim

plif

y.

L ≈

424.1

cm

2;

L ≈

157.1

ft2

;

S ≈

678.6

cm

2

S

≈ 2

35.6

ft2

L ≈

204.2

cm

2;

L ≈

71.1

in

2;

S ≈

282.7

cm

2

S

≈ 1

21.4

in

2

001_

025_

GE

OC

RM

C12

_890

521.

indd

193/

27/1

010

:45:

38P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

2

0

Gle

ncoe

Geo

met

ry

12-3

Skill

s Pr

acti

ceS

urf

ace A

reas o

f P

yra

mid

s a

nd

Co

nes

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

reg

ula

r p

yram

id. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

2.

3.

4.

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

con

e. R

oun

d t

o th

e n

eare

st t

enth

.

5.

6.

7.

8.

4 cm

7 cm

20 in

.

8 in

.

9 m 10

m14

ft

12 ft

L

= 5

6 c

m2

L =

480 i

n2

S

= 7

2 c

m2

S =

646.3

in

2

L

≈ 283.2

m2

L ≈

389.0

ft2

S

≈ 4

55.3

m2

S ≈

585.0

ft2

14 m

5 m

10 ft

25 ft

8 in

.

21 in

.

17 m

m9 m

m

L

≈ 219.9

m2

L ≈

845.9

ft2

S

≈ 2

98.5

m2

S ≈

1160.1

ft2

L

≈ 5

27.8

in

2

L ≈

480.7

mm

2

S

≈ 7

28.8

in

2

S ≈

735.1

mm

2

001_

025_

GE

OC

RM

C12

_890

521.

indd

206/

14/0

82:

46:2

9P

M


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-3

Cha

pte

r 12

21

Gle

ncoe

Geo

met

ry

12-3

Prac

tice

S

urf

ace A

reas o

f P

yra

mid

s a

nd

Co

nes

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

reg

ula

r p

yram

id. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

9 yd

10 y

d

2.

12 m

7 m

3.

13 ft

5 ft

4.

8 cm

2.5

cm

Fin

d t

he

late

ral

area

an

d s

urf

ace

area

of

each

con

e. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

5.

5 m

4 m

6.

7 cm

21 c

m

7. F

ind

the

surf

ace

area

of

a co

ne

if t

he

hei

ght

is 1

4 ce

nti

met

ers

and

the

slan

t h

eigh

t is

16

.4 c

enti

met

ers.

8. F

ind

the

surf

ace

area

of

a co

ne

if t

he

hei

ght

is 1

2 in

ches

an

d th

e di

amet

er i

s 27

in

ches

.

9. G

AZE

BO

S T

he

roof

of

a ga

zebo

is

a re

gula

r oc

tago

nal

pyr

amid

. If

the

base

of

the

pyra

mid

has

sid

es o

f 0.

5 m

eter

an

d th

e sl

ant

hei

ght

of t

he

roof

is

1.9

met

ers,

fin

d th

e ar

ea o

f th

e ro

of.

10. H

ATS

Cuo

ng b

ough

t a

coni

cal

hat

on a

rec

ent

trip

to

cent

ral

Vie

tnam

. The

bas

ic f

ram

e of

th

e h

at i

s 16

hoo

ps o

f ba

mbo

o th

at g

radu

ally

dim

inis

h i

n s

ize.

Th

e h

at i

s co

vere

d in

pa

lm l

eave

s. I

f th

e h

at h

as a

dia

met

er o

f 50

cen

tim

eter

s an

d a

slan

t h

eigh

t of

32

cen

tim

eter

s, w

hat

is

the

late

ral

area

of

the

con

ical

hat

?

L =

180 y

d2;

S =

261 y

d2

L =

126 m

2; S

≈ 147.2

m2

L =

162.5

ft2

; S

≈ 2

05.5

ft2

L ≈

60 c

m2;

S ≈

76.2

cm

2

L ≈

80.5

m2;

S ≈

130.7

cm

2

468.8

cm

2 ; S

≈ 640.7

cm

2

669.3

cm

2

1338.6

in

2

3.8

m2

ab

ou

t 2513.3

cm

2

001_

025_

GE

OC

RM

C12

_890

521.

indd

215/

30/0

93:

07:3

4P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

2

2

Gle

ncoe

Geo

met

ry

12-3

Wor

d Pr

oble

m P

ract

ice

Su

rface A

reas o

f P

yra

mid

s a

nd

Co

nes

1. P

APE

R M

OD

ELS

Pat

rick

is

mak

ing

a pa

per

mod

el

of a

cas

tle.

Par

t of

th

e m

odel

in

volv

es

cutt

ing

out

the

net

sh

own

an

d fo

ldin

g it

in

to a

py

ram

id. T

he

pyra

mid

has

a s

quar

e ba

se. W

hat

is

the

late

ral

surf

ace

area

of

the

resu

ltin

g py

ram

id?

2. T

ETR

AH

EDR

ON

Su

ng

Li

buil

ds a

pap

er

mod

el o

f a

regu

lar

tetr

ahed

ron

, a

pyra

mid

wit

h a

n e

quil

ater

al t

rian

gle

for

the

base

an

d th

ree

equ

ilat

eral

tri

angl

es

for

the

late

ral

face

s. O

ne

of t

he

face

s of

th

e te

trah

edro

n h

as a

n a

rea

of 1

7 sq

uar

e in

ches

. Wh

at i

s th

e to

tal

surf

ace

area

of

the

tetr

ahed

ron

?

3. P

APE

RW

EIG

HTS

Dap

hn

e u

ses

a pa

perw

eigh

t sh

aped

lik

e a

pyra

mid

wit

h

a re

gula

r h

exag

on f

or a

bas

e. T

he

side

le

ngt

h o

f th

e re

gula

r h

exag

on i

s 1

inch

. T

he

alti

tude

of

the

pyra

mid

is

2 in

ches

.

W

hat

is

the

late

ral

surf

ace

area

of

this

py

ram

id?

Rou

nd

you

r an

swer

s to

th

e n

eare

st h

un

dred

th.

4. S

PRA

Y P

AIN

T A

can

of

spra

y pa

int

shoo

ts o

ut

pain

t in

a c

one

shap

ed m

ist.

T

he

late

ral

surf

ace

area

of

the

con

e is

65

π sq

uar

e in

ches

wh

en t

he

can

is

hel

d 12

in

ches

fro

m a

can

vas.

Wh

at i

s th

e ar

ea o

f th

e pa

rt o

f th

e ca

nva

s th

at g

ets

spra

yed

wit

h p

ain

t? R

oun

d yo

ur

answ

er

to t

he

nea

rest

hu

ndr

edth

.

5. M

EGA

PHO

NES

A m

egap

hon

e is

fo

rmed

by

taki

ng

a co

ne

wit

h a

rad

ius

of 2

0 ce

nti

met

ers

and

an a

ltit

ude

of

60 c

enti

met

ers

and

cutt

ing

off

the

tip.

T

he

cut

is m

ade

alon

g a

plan

e th

at i

s pe

rpen

dicu

lar

to t

he

axis

of

the

con

e an

d in

ters

ects

th

e ax

is 1

2 ce

nti

met

ers

from

th

e ve

rtex

. Rou

nd

you

r an

swer

s to

th

e n

eare

st h

un

dred

th.

a. W

hat

is

the

late

ral

surf

ace

area

of

the

orig

inal

con

e?

b.

Wh

at i

s th

e la

tera

l su

rfac

e ar

ea o

f th

e ti

p th

at i

s re

mov

ed?

c. W

hat

is

the

late

ral

surf

ace

area

of

the

meg

aph

one?

20 c

m20

cm

15 c

m

600 c

m2

68 i

n2

6.5

4 i

n2

3973.8

4 i

n2

158.9

5 i

n2

3814.8

9 i

n2

78.5

4 i

n2

001_

025_

GE

OC

RM

C12

_890

521.

indd

224/

10/0

89:

14:3

5P

M

Lesson 12-3


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

23

Gle

ncoe

Geo

met

ry

Co

ne P

att

ern

s

Th

e p

atte

rn a

t th

e ri

ght

is m

ade

from

a

circ

le. I

t ca

n b

e fo

lded

to

mak

e a

con

e.

1. M

easu

re t

he

radi

us

of t

he

circ

le t

o th

e n

eare

st c

enti

met

er.

2. T

he

patt

ern

is

wh

at f

ract

ion

of

the

com

plet

e ci

rcle

?

3. W

hat

is

the

circ

um

fere

nce

of

the

com

plet

e ci

rcle

?

4. H

ow l

ong

is t

he

circ

ula

r ar

c th

at i

s th

e ou

tsid

e of

th

e pa

tter

n?

5. C

ut

out

the

patt

ern

an

d ta

pe i

t to

geth

er t

o fo

rm a

con

e.

6. M

easu

re t

he

diam

eter

of

the

circ

ula

r ba

se o

f th

e co

ne.

7. W

hat

is

the

circ

um

fere

nce

of

the

base

of

the

con

e?

8. W

hat

is

the

slan

t h

eigh

t of

th

e co

ne?

9. U

se t

he

Pyt

hag

orea

n T

heo

rem

to

calc

ula

te t

he

hei

ght

of t

he

con

e.

Use

a d

ecim

al a

ppro

xim

atio

n. C

hec

k yo

ur

calc

ula

tion

by

mea

suri

ng

the

hei

ght

wit

h a

met

ric

rule

r.

10. F

ind

the

late

ral

area

.

11. F

ind

the

tota

l su

rfac

e ar

ea.

Mak

e a

pap

er p

atte

rn f

or e

ach

con

e w

ith

th

e gi

ven

mea

sure

men

ts.

Th

en c

ut

the

pat

tern

ou

t an

d m

ake

the

con

e. F

ind

th

e m

easu

rem

ents

.

12.

120°

6 cm

13

.

20 c

m

d

iam

eter

of

base

=

dia

met

er o

f ba

se =

l

ater

al a

rea

=

lat

eral

are

a =

h

eigh

t of

con

e =

h

eigh

t of

con

e =

(to

nea

rest

ten

th o

f a

cen

tim

eter

) (

to n

eare

st t

enth

of

a ce

nti

met

er)

Enri

chm

ent

12-3

2 c

m

3

−

4

4π

cm

3π

cm

See s

tud

en

ts’

wo

rk.

3π

cm

2 c

m

3 c

m

1.3

2 c

m

3π

cm

2

5.2

5 c

m2

8 c

m10 c

m

24

π c

m2

50

π c

m2

4.5

cm

8.7

cm

001_

025_

GE

OC

RM

C12

_890

521.

indd

235/

30/0

94:

13:2

3P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

2

4

Gle

ncoe

Geo

met

ry

12-3

You

can

use

a s

prea

dsh

eet

to d

eter

min

e th

e su

rfac

e ar

ea o

f a

con

e.

L

ucy

wan

ts t

o w

rap

a M

oth

er’s

Day

gif

t. T

he

gift

sh

e h

as b

ough

t fo

r h

er m

oth

er i

s in

a c

onic

al b

ox t

hat

has

a s

lan

t h

eigh

t of

6 i

nch

es a

nd

has

a r

adiu

s of

3 i

nch

es. S

he

mu

st d

eter

min

e th

e su

rfac

e ar

ea o

f th

e b

ox t

o d

eter

min

e h

ow

mu

ch w

rap

pin

g p

aper

to

bu

y. U

se a

sp

read

shee

t to

det

erm

ine

the

surf

ace

area

of

the

box

. Rou

nd

to

the

nea

rest

ten

th.

Ste

p 1

U

se c

ell

A1

for

the

radi

us

of t

he

con

e an

d ce

ll B

1 fo

r th

e h

eigh

t.

Ste

p 2

In

cel

l C

1, e

nte

r an

equ

als

sign

fol

low

ed b

y P

I()*

A1*

B1

+ P

I()*

A1^

2. T

hen

pre

ss

EN

TE

R. T

his

wil

l re

turn

th

e su

rfac

e ar

ea o

f th

e co

ne.

Th

e su

rfac

e ar

ea o

f th

e co

nic

al b

ox i

s 84

.8 i

n2

to t

he

nea

rest

ten

th.

U

se a

sp

read

shee

t to

det

erm

ine

the

surf

ace

area

of

a co

ne

that

has

a r

adiu

s of

2.

5 ce

nti

met

ers

and

a s

lan

t h

eigh

t of

5.2

cen

tim

eter

s.

Rou

nd

to

the

nea

rest

ten

th.

Ste

p 1

U

se c

ell

A2

for

the

radi

us

of t

he

con

e an

d ce

ll B

2 fo

r th

e sl

ant

hei

ght.

Ste

p 2

C

lick

on

th

e bo

ttom

rig

ht

corn

er o

f ce

ll C

1 an

d dr

ag i

t to

C2.

Th

is r

etu

rns

the

surf

ace

area

of

the

con

e.

Th

e su

rfac

e ar

ea o

f th

e co

ne

is 6

0.5

cm2

to t

he

nea

rest

ten

th.

Exer

cise

sU

se a

sp

read

shee

t to

fin

d t

he

surf

ace

area

of

each

con

e w

ith

th

e gi

ven

d

imen

sion

s. R

oun

d t

o th

e n

eare

st t

enth

.

1. r

= 1

2 m

, � =

2.3

m

2.

r =

6 m

, � =

2 m

3. r

= 3

in

., � =

7 i

n.

4.

r =

5 i

n.,

� =

11

in.

5. r

= 1

ft,

� =

3 f

t

6. r

= 3

ft,

� =

1.5

ft

7. r

= 1

0 m

m, �

= 2

0 m

m

8. r

= 1

.5 m

m, �

= 4

.5 m

m

9. r

= 6

.2 c

m, �

= 1

.2 c

m

10. r

= 1

0 cm

, � =

15

cm

11. r

= 1

0 m

, � =

2 m

12

. r =

11

m, �

= 1

3 m

Spre

adsh

eet

Act

ivit

yS

urf

ace A

reas o

f C

on

es

Exam

ple

1

Exam

ple

2

A1 2

BC

Sh

eet

1

539.1

m2

94.2

in

2

12.6

ft2 9

42.5

mm

2

144.1

cm

2

377.0

m2

150.8

m2

251.3

in

2

42.4

ft2

28.3

mm

2

785.4

cm

2

829.4

m2

001_

025_

GE

OC

RM

C12

_890

521.

indd

245/

30/0

93:

07:4

7P

M


NA

ME

DA

TE

PE

RIO

D

NA

ME

DA

TE

PE

RIO

D

Lesson 12-4

Cha

pte

r 12

25

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

Vo

lum

es o

f P

rism

s a

nd

Cylin

ders

Vo

lum

es o

f Pr

ism

s T

he

mea

sure

of

the

amou

nt

of s

pace

th

at a

th

ree-

dim

ensi

onal

fig

ure

en

clos

es i

s th

e vo

lum

e of

th

e fi

gure

. Vol

um

e is

mea

sure

d in

un

its

such

as

cubi

c fe

et, c

ubi

c ya

rds,

or

cubi

c m

eter

s. O

ne

cubi

c u

nit

is

the

volu

me

of a

cu

be

that

mea

sure

s on

e u

nit

on

eac

h e

dge.

Vo

lum

e

of

a P

ris

m

If a

prism

ha

s a

vo

lum

e o

f V

cu

bic

un

its,

a h

eig

ht

of

h u

nits,

an

d e

ach

ba

se

ha

s a

n a

rea

of

B s

qu

are

un

its,

the

n V

= B

h.

F

ind

th

e vo

lum

e of

th

e p

rism

.

7 cm

3 cm4

cm

V =

Bh

V

olu

me o

f a p

rism

=

(7)

(3)(

4)

B =

(7)(

3),

h =

4

=

84

Multip

ly.

Th

e vo

lum

e of

th

e pr

ism

is

84 c

ubi

c ce

nti

met

ers.

F

ind

th

e vo

lum

e of

th

e p

rism

if

the

area

of

each

bas

e is

6.3

sq

uar

e fe

et. 3.

5 ft

base

V =

Bh

V

olu

me o

f a p

rism

=

(6.

3)(3

.5)

B =

6.3

, h

= 3

.5

=

22.

05

Multip

ly.

Th

e vo

lum

e is

22.

05 c

ubi

c fe

et.

Exer

cise

s

Fin

d t

he

volu

me

of e

ach

pri

sm.

1.

8 ft

8 ft

8 ft

2.

3 cm

4 cm

1.5

cm

3.

30°

15 ft12

ft

4.

10 ft

15 ft

12 ft

5.

4 cm

6 cm

2 cm

1.5

cm

6.

7 yd

4 yd

3 yd

12-4

Exam

ple

1Ex

amp

le 2

cubi

c fo

otcu

bic

yard

27 c

ubic

feet

= 1

cubic

yard

512 f

t39 c

m3

467.7

ft3

1800 f

t3

27 c

m3

84 y

d3

001_

025_

GE

OC

RM

C12

_890

521.

indd

254/

10/0

89:

14:4

7P

M

Answers (Lesson 12-3 and Lesson 12-4)


Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

2

6

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Vo

lum

es o

f P

rism

s a

nd

Cylin

ders

Vo

lum

es o

f C

ylin

der

s T

he

volu

me

of a

cyl

inde

r is

th

e pr

odu

ct o

f th

e

hei

ght

and

the

area

of

the

base

. Wh

en a

sol

id i

s n

ot a

rig

ht

soli

d, u

se

Cav

alie

ri’s

Pri

nic

iple

to

fin

d th

e vo

lum

e. T

he

prin

cipl

e st

ates

th

at i

f tw

o so

lids

hav

e th

e sa

me

hei

ght

and

the

sam

e cr

oss

sect

ion

al a

rea

at e

very

le

vel,

then

th

ey h

ave

the

sam

e vo

lum

e.

F

ind

th

e vo

lum

e of

th

e cy

lin

der

.

4 cm3 cm

V =

πr2 h

V

olu

me o

f a c

ylin

der

=

π(3

)2 (4)

r

= 3

, h

= 4

≈

113

.1

Sim

plif

y.

Th

e vo

lum

e is

abo

ut

113.

1 cu

bic

cen

tim

eter

s.

F

ind

th

e vo

lum

e of

th

e ob

liq

ue

cyli

nd

er.

8 in

.13 in

.

5 in

.

h Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

the

hei

ght

of

the

cyli

nde

r.h

2 +

52

= 1

32 P

yth

agore

an T

heore

m

h

2 =

144

S

implif

y.

h

= 1

2 T

ake t

he s

quare

root

of

each s

ide.

V

= π

r2 h

Volu

me o

f a c

ylin

der

=

π(4

)2 (12

) r

= 4

, h

= 1

2

≈

603

.2

Sim

plif

y.

Th

e V

olu

me

is a

bou

t 60

3.2

cubi

c in

ches

.Ex

erci

ses

Fin

d t

he

volu

me

of e

ach

cyl

ind

er. R

oun

d t

o th

e n

eare

st t

enth

.

1.

2 ft

1 ft

2.

18

cm

2 cm

1

2.6

ft3

226.2

cm

3

3.

12 ft

1.5

ft

4.

20

ft

20 ft

8

4.8

ft3

6

283.2

ft3

5.

10 c

m

13 c

m

6.

1 yd

4 yd

6

52.4

cm

3

12.6

yd

3

12-4

Vo

lum

e o

f

a C

yli

nd

er

If a

cylin

de

r h

as a

vo

lum

e o

f V

cu

bic

un

its,

a h

eig

ht

of

h u

nits,

an

d t

he

ba

se

s h

ave

a r

ad

ius o

f r

un

its,

the

n V

= π

r2h.

Exam

ple

1Ex

amp

le 2

r

h

026_

044_

GE

OC

RM

C12

_890

521.

indd

264/

14/0

89:

21:1

8P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-4

Cha

pte

r 12

27

Gle

ncoe

Geo

met

ry

12-4

Fin

d t

he

volu

me

of e

ach

pri

sm o

r cy

lin

der

. Rou

nd

to

the

nea

rest

ten

th

if n

eces

sary

.

1.

18 c

m

16 c

m8 cm

2.

6 ft

8 ft

2 ft

2

304 c

m3

96 f

t3

3.

3 m

5 m

13 m

4.

16 in

.22

in.

34 in

.

9

0 m

3

5280 i

n3

5.

15 m

m23

mm

6.

6

yd 10 y

d

1

6,2

57.7

mm

3

226.2

yd

3

Fin

d t

he

volu

me

of e

ach

ob

liq

ue

pri

sm o

r cy

lin

der

. Rou

nd

to

the

nea

rest

te

nth

if

nec

essa

ry.

7.

8.

5 in

.

3 in

.

1224 c

m3

141.4

in

3

Skill

s Pr

acti

ceV

olu

mes o

f P

rism

s a

nd

Cylin

ders

17 c

m

18 c

m

4 cm

026_

044_

GE

OC

RM

C12

_890

521.

indd

276/

14/0

82:

47:1

7P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

2

8

Gle

ncoe

Geo

met

ry

Prac

tice

Vo

lum

es o

f P

rism

s a

nd

Cylin

ders

Fin

d t

he

volu

me

of e

ach

pri

sm o

r cy

lin

der

. Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.

17 m

10 m

26 m

2.

5 in

.

5 in

.

5 in

.

9 in

.

2

040 m

3

97.4

in

3

3. 16

mm

17.5

mm

4.

7 ft

25 ft

3

518.6

mm

3

923.6

ft3

5.

13 y

d

20 y

d

10 y

d

6. 30

cm

8 cm

2

600 y

d3

6031.9

cm

3

7. A

QU

AR

IUM

Mr.

Gu

tier

rez

purc

has

ed a

cyl

indr

ical

aqu

ariu

m f

or h

is o

ffic

e.

Th

e aq

uar

ium

has

a h

eigh

t of

25

1 −

2 i

nch

es a

nd

a ra

diu

s of

21

inch

es.

a. W

hat

is

the

volu

me

of t

he

aqu

ariu

m i

n c

ubi

c fe

et?

20.4

ft3

b.

If t

her

e ar

e 7.

48 g

allo

ns

in a

cu

bic

foot

, how

man

y ga

llon

s of

wat

er d

oes

the

aqu

ariu

m

hol

d?

152.9

gal

c. I

f a

cubi

c fo

ot o

f w

ater

wei

ghs

abou

t 62

.4 p

oun

ds, w

hat

is

the

wei

ght

of t

he

wat

er i

n

the

aqu

ariu

m t

o th

e n

eare

st f

ive

pou

nds

?

1275 l

b

12-4

026_

044_

GE

OC

RM

C12

_890

521.

indd

284/

14/0

811

:10:

42A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-4

Cha

pte

r 12

29

Gle

ncoe

Geo

met

ry

Wor

d Pr

oble

m P

ract

ice

Vo

lum

es o

f P

rism

s a

nd

Cylin

ders

1. T

RA

SH C

AN

S T

he

Mey

er f

amil

y u

ses

a ki

tch

en t

rash

can

sh

aped

lik

e a

cyli

nde

r.

It h

as a

hei

ght

of

18 i

nch

es a

nd

a ba

se

diam

eter

of

12 i

nch

es.

Wh

at i

s th

e vo

lum

e of

th

e tr

ash

can

? R

oun

d yo

ur

answ

er t

o th

e n

eare

st t

enth

of

a cu

bic

inch

.

2035.8

in

3

2. B

ENC

H I

nsi

de a

lob

by, t

her

e is

a p

iece

of

fu

rnit

ure

for

sit

tin

g. T

he

furn

itu

re i

s sh

aped

lik

e a

sim

ple

bloc

k w

ith

a s

quar

e ba

se 6

fee

t on

eac

h s

ide

and

a h

eigh

t of

1

3 − 5

feet

.

6 ft

6 ft

1ft

3 5

Wh

at i

s th

e vo

lum

e of

th

e se

at?

5

7.6

ft3

3. F

RA

MES

Mar

gare

t m

akes

a s

quar

e fr

ame

out

of f

our

piec

es o

f w

ood.

Eac

h

piec

e of

woo

d is

a

rect

angu

lar

pris

m

wit

h a

len

gth

of

40 c

enti

met

ers,

a

hei

ght

of

4 ce

nti

met

ers,

an

d a

dept

h o

f 6

cen

tim

eter

s.

Wh

at i

s th

e to

tal

volu

me

of t

he

woo

d u

sed

in t

he

fram

e?

3

840 c

m3

4. P

ENC

IL G

RIP

S A

pen

cil

grip

is

shap

ed

like

a t

rian

gula

r pr

ism

wit

h a

cyl

inde

r re

mov

ed f

rom

th

e m

iddl

e. T

he

base

of

the

pris

m i

s a

righ

t is

osce

les

tria

ngl

e w

ith

leg

len

gth

s of

2 c

enti

met

ers.

Th

e di

amet

er o

f th

e ba

se o

f th

e re

mov

ed

cyli

nde

r is

1 c

enti

met

er. T

he

hei

ghts

of

the

pris

m a

nd

the

cyli

nde

r ar

e th

e sa

me,

an

d eq

ual

to

4 ce

nti

met

ers.

Wh

at i

s th

e ex

act

volu

me

of t

he

pen

cil

grip

?

8

- π

cm

3

5. T

UN

NEL

S C

onst

ruct

ion

wor

kers

are

di

ggin

g a

tun

nel

th

rou

gh a

mou

nta

in.

Th

e sp

ace

insi

de t

he

tun

nel

is

goin

g to

be

sh

aped

lik

e a

rect

angu

lar

pris

m. T

he

mou

th o

f th

e tu

nn

el w

ill

be a

rec

tan

gle

20 f

eet

hig

h a

nd

50 f

eet

wid

e an

d th

e le

ngt

h o

f th

e tu

nn

el w

ill

be 9

00 f

eet.

a. W

hat

will

the

vol

ume

of t

he t

unne

l be?

900,0

00 f

t3

b.

If i

nst

ead

of a

rec

tan

gula

r sh

ape,

th

e tu

nn

el h

ad a

sem

icir

cula

r sh

ape

wit

h

a 50

-foo

t di

amet

er, w

hat

wou

ld b

e it

s vo

lum

e? R

oun

d yo

ur

answ

er t

o th

e n

eare

st c

ubi

c fo

ot.

883,5

73 f

t3

18 in

.

12 in

.

12-4

026_

044_

GE

OC

RM

C12

_890

521.

indd

294/

10/0

89:

11:4

1P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

3

0

Gle

ncoe

Geo

met

ry

Vis

ible

Su

rface A

rea

Use

pap

er, s

ciss

ors,

an

d t

ape

to m

ake

five

cu

bes

th

at h

ave

one-

inch

ed

ges.

A

rran

ge t

he

cub

es t

o fo

rm e

ach

sh

ape

show

n. T

hen

fin

d t

he

volu

me

and

th

e vi

sib

le s

urf

ace

area

. In

oth

er w

ord

s, d

o n

ot i

ncl

ud

e th

e ar

ea o

f su

rfac

e co

vere

d b

y ot

her

cu

bes

or

by

the

tab

le o

r d

esk

.

1.

2.

v

olu

me

= 4 i

n3

vol

um

e =

4 i

n3

v

isib

le s

urf

ace

area

= 14 i

n2

vis

ible

su

rfac

e ar

ea =

15 i

n2

3.

4.

5.

v

olu

me

= 5 i

n3

vol

um

e =

5 i

n3

vol

um

e =

5 i

n3

v

isib

le s

urfa

ce a

rea

= 1

7 in

2

vis

ible

sur

face

are

a =

19 in

2

vis

ible

sur

face

are

a =

19 in

2

6. F

ind

the

volu

me

and

the

visi

ble

surf

ace

ar

ea o

f th

e fi

gure

at

the

righ

t.

v

olu

me

= 136 i

n3

v

isib

le s

urf

ace

area

= 164 i

n2

4 in

.

4 in

.

3 in

.

3 in

.

3 in

.

8 in

.

3 in

. 5 in

.

5 in

.

12-4

Enri

chm

ent

026_

044_

GE

OC

RM

C12

_890

521.

indd

304/

11/0

86:

34:3

7A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-5

Cha

pte

r 12

31

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

Vo

lum

es o

f P

yra

mid

s a

nd

Co

nes

Vo

lum

es o

f Py

ram

ids

Th

is f

igu

re s

how

s a

pris

m a

nd

a py

ram

id

that

hav

e th

e sa

me

base

an

d th

e sa

me

hei

ght.

It

is c

lear

th

at t

he

volu

me

of t

he

pyra

mid

is

less

th

an t

he

volu

me

of t

he

pris

m. M

ore

spec

ific

ally

, th

e vo

lum

e of

th

e py

ram

id i

s on

e-th

ird

of t

he

volu

me

of t

he

pris

m.

F

ind

th

e vo

lum

e of

th

e sq

uar

e p

yram

id.

V =

1 −

3 B

h

Volu

me o

f a p

yra

mid

=

1 −

3 (8

)(8)

10

B =

(8)(

8),

h =

10

≈

213

.3

Multip

ly.

Th

e vo

lum

e is

abo

ut

213.

3 cu

bic

feet

.

Exer

cise

sF

ind

th

e vo

lum

e of

eac

h p

yram

id. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

12 ft

8 ft

10 ft

2.

10 ft

6 ft

15 ft

3

20 f

t3

120 f

t3

3.

4 cm

8 cm

12 c

m

4.

18 ft

regu

lar

hexa

gon

6 ft

1

10.9

cm

3

561.2

ft3

5.

15 in

.

15 in

.

16 in

.

6.

6 yd

8 yd

5 yd

1

200 i

n3

64 y

d3

8 ft

8 ft

10 ft

12-5

Vo

lum

e o

f

a P

yra

mid

If a

pyra

mid

ha

s a

vo

lum

e o

f V

cu

bic

un

its,

a h

eig

ht

of

h u

nits,

an

d a

ba

se

with

an

are

a o

f B

sq

ua

re u

nits,

the

n V

= 1

−

3 B

h.

Exam

ple

026_

044_

GE

OC

RM

C12

_890

521.

indd

314/

10/0

89:

11:4

7P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

3

2

Gle

ncoe

Geo

met

ry

Vo

lum

es o

f C

on

es F

or a

con

e, t

he

volu

me

is o

ne-

thir

d th

e pr

odu

ct o

f th

e

hei

ght

and

the

area

of

the

base

. Th

e ba

se o

f a

con

e is

a c

ircl

e, s

o th

e ar

ea o

f th

e ba

se i

s π

r2 .

F

ind

th

e vo

lum

e of

th

e co

ne.

V

= 1 −

3 π

r2 h

Volu

me o

f a c

one

=

1 −

3 π

(5)2 1

2 r

= 5

, h

= 1

2

≈

314

.2

Sim

plif

y.

Th

e vo

lum

e of

th

e co

ne

is a

bou

t 31

4.2

cubi

c ce

nti

met

ers.

Exer

cise

sF

ind

th

e vo

lum

e of

eac

h c

one.

Rou

nd

to

the

nea

rest

ten

th.

1.

6 cm

10 c

m

2.

8 ft

10 ft

3

01.6

cm

3

670.2

ft3

3.

30 in

.

12 in

.

4.

45

°18

yd 20

yd

1

131.0

in

3

1332.9

yd

3

5.

26 ft

20 ft

6.

16 c

m

45°

2

513.3

ft3

3

79.1

cm

3

12 c

m

5 cm

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Vo

lum

es o

f P

yra

mid

s a

nd

Co

nes

12-5 V

olu

me

of

a C

on

e

If a

co

ne

ha

s a

vo

lum

e o

f V

cu

bic

un

its,

a h

eig

ht

of

h u

nits,

an

d t

he

ba

se

s h

ave

a r

ad

ius o

f r

un

its,

the

n V

= 1

−

3 π

r2 h.

Exam

ple

r

h

026_

044_

GE

OC

RM

C12

_890

521.

indd

324/

10/0

89:

11:5

2P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-5

Cha

pte

r 12

3

3

Gle

ncoe

Geo

met

ry

12-5

Skill

s Pr

acti

ceV

olu

mes o

f P

yra

mid

s a

nd

Co

nes

Fin

d t

he

volu

me

of e

ach

pyr

amid

or

con

e. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

2.

6

6.7

ft3

7

4.7

cm

3

3.

4.

3

57.8

in

3

3769.9

m3

5.

6.

1

231.5

yd

3

1210.6

mm

3

Fin

d t

he

volu

me

of e

ach

ob

liq

ue

pyr

amid

or

con

e. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

7.

8.

31 f

t3

452.4

cm

3

5 ft

5 ft

8 ft

4 cm

7 cm

8 cm

8 in

.10

in.

14 in

.25

m12 m

25 y

d

14 y

d66

°

18 m

m

4 ft

4 ft

6 ft

12 c

m6 cm

026_

044_

GE

OC

RM

C12

_890

521.

indd

336/

14/0

82:

47:3

5P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

3

4

Gle

ncoe

Geo

met

ry

Prac

tice

Vo

lum

es o

f P

yra

mid

s a

nd

Co

nes

Fin

d t

he

volu

me

of e

ach

pyr

amid

or

con

e. R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.

9.2

yd9.

2 yd

13 y

d

2.

12.5

cm

25 c

m

23 c

m

3

17.5

yd

3

2395.8

cm

3

3.

19 ft

9 ft

4.

52°

12 m

m

1

419.4

ft3

1

104.6

mm

3

5.

6 in

.6

in.

11 in

. 6.

37 ft

11 ft

1

32 i

n3

4688.3

ft3

7. C

ON

STR

UC

TIO

N M

r. G

anty

bu

ilt

a co

nic

al s

tora

ge s

hed

. Th

e ba

se o

f th

e sh

ed i

s 4

met

ers

in d

iam

eter

an

d th

e h

eigh

t of

th

e sh

ed i

s 3.

8 m

eter

s. W

hat

is

the

volu

me

of

the

shed

?

a

bo

ut

15.9

m3

8. H

ISTO

RY

Th

e st

art

of t

he

pyra

mid

age

beg

an w

ith

Kin

g Z

oser

’s p

yram

id, e

rect

ed i

n t

he

27th

cen

tury

B.C

. In

its

ori

gin

al s

tate

, it

stoo

d 62

met

ers

hig

h w

ith

a r

ecta

ngu

lar

base

th

at m

easu

red

140

met

ers

by 1

18 m

eter

s. F

ind

the

volu

me

of t

he

orig

inal

pyr

amid

.

a

bo

ut

341,4

13.3

m3

12-5

026_

044_

GE

OC

RM

C12

_890

521.

indd

344/

10/0

89:

12:0

0P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-5

Cha

pte

r 12

35

Gle

ncoe

Geo

met

ry

1. I

CE

CR

EAM

DIS

HES

Th

e pa

rt o

f a

dish

de

sign

ed f

or i

ce c

ream

is

shap

ed l

ike

an

ups

ide-

dow

n c

one.

Th

e ba

se o

f th

e co

ne

has

a r

adiu

s of

2 i

nch

es a

nd

the

hei

ght

is 1

.2 i

nch

es.

Wh

at i

s th

e vo

lum

e of

th

e co

ne?

Rou

nd

you

r an

swer

to

the

nea

rest

hu

ndr

edth

.

5

.03 i

n3

2. G

REE

NH

OU

SES

A g

reen

hou

se h

as t

he

shap

e of

a s

quar

e py

ram

id. T

he

base

has

a

side

len

gth

of

30 y

ards

. Th

e h

eigh

t of

th

e gr

een

hou

se i

s 18

yar

ds.

18 yd

30

yd

Wh

at i

s th

e vo

lum

e of

th

e gr

een

hou

se?

5

400 y

d3

3. T

EEPE

E C

aitl

yn m

ade

a te

epee

for

a

clas

s pr

ojec

t. H

er t

eepe

e h

ad a

dia

met

er

of 6

fee

t. T

he

angl

e th

e si

de o

f th

e te

epee

m

ade

wit

h t

he

grou

nd

was

65°

.

65˚

Wh

at w

as t

he

volu

me

of t

he

teep

ee?

Rou

nd

you

r an

swer

to

the

nea

rest

h

un

dred

th.

6

0.6

3 f

t3

4. S

CU

LPTI

NG

A s

culp

tor

wan

ts t

o re

mov

e st

one

from

a c

ylin

dric

al b

lock

3 f

eet

hig

h

and

turn

it

into

a c

one.

Th

e di

amet

er o

f th

e ba

se o

f th

e co

ne

and

cyli

nde

r is

2

feet

.

Wh

at i

s th

e vo

lum

e of

th

e st

one

that

th

e sc

ulp

tor

mu

st r

emov

e? R

oun

d yo

ur

answ

er t

o th

e n

eare

st h

un

dred

th.

6

.28 f

t3

5. S

TAG

ESA

sta

ge h

as t

he

form

of

a sq

uar

e py

ram

id w

ith

th

e to

p sl

iced

off

al

ong

a pl

ane

para

llel

to

the

base

. Th

e si

de l

engt

h o

f th

e to

p sq

uar

e is

12

feet

an

d th

e si

de l

engt

h o

f th

e bo

ttom

squ

are

is 1

6 fe

et. T

he

hei

ght

of t

he

stag

e is

3

feet

.

12

feet

16

feet

3 fe

et

a. W

hat

is

the

volu

me

of t

he

enti

re

squ

are

pyra

mid

th

at t

he

stag

e is

pa

rt o

f?

1024 f

t3

b.

Wh

at i

s th

e vo

lum

e of

th

e to

p of

th

e py

ram

id t

hat

is

rem

oved

to

get

the

stag

e?

432 f

t3

c. W

hat

is

the

volu

me

of t

he

stag

e?

592 f

t3

Wor

d Pr

oble

m P

ract

ice

Vo

lum

es o

f P

yra

mid

s a

nd

Co

nes

12-5

026_

044_

GE

OC

RM

C12

_890

521.

indd

355/

30/0

93:

08:5

1P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

3

6

Gle

ncoe

Geo

met

ry

Enri

chm

ent

Fru

stu

ms

A f

rust

um

is

a fi

gure

for

med

wh

en a

pla

ne

inte

rsec

ts a

pyr

amid

or

con

e so

th

at t

he

plan

e is

par

alle

l to

th

e so

lid’

s ba

se. T

he

fru

stu

m i

s th

e pa

rt o

f th

e so

lid

betw

een

th

e pl

ane

and

the

base

. To

fin

d th

e vo

lum

e of

a f

rust

um

, th

e ar

eas

of b

oth

bas

es m

ust

be

calc

ula

ted

and

use

d in

th

e fo

rmu

la.

V =

1 −

3 h

(B1

+ B

2 +

√ �

�

B1B

2 ),w

her

e h

= h

eigh

t (p

erpe

ndi

cula

r di

stan

ce b

etw

een

th

e ba

ses)

,B

1 =

are

a of

top

bas

e, a

nd

B2

= a

rea

of b

otto

m b

ase.

Des

crib

e th

e sh

ape

of t

he

bas

es o

f ea

ch f

rust

um

. Th

en f

ind

th

e vo

lum

e. R

oun

d t

o th

e n

eare

st t

enth

.

1.

13 c

m 6 cm

9 cm

5 cm

19.5

cm

2.

7.5

in.

4.5

in.

3 in

.

r

ecta

ng

les;

617.5

cm

3

cir

cle

s;

335.8

in

3

3.

8 m

6 m

12 m

4.5

m2.

25 m3

m

5 m

4.

12 ft

13 ft

7 ft

tra

pezo

ids;

151.6

m3

cir

cle

s;

3480.9

ft3

12-5

026_

044_

GE

OC

RM

C12

_890

521.

indd

366/

1/09

10:1

9:21

AM


NA

ME

DA

TE

PE

RIO

D

Lesson 12-6

Cha

pte

r 12

37

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

Su

rface A

reas a

nd

Vo

lum

es o

f S

ph

ere

s

Surf

ace

Are

as o

f Sp

her

es Y

ou c

an t

hin

k of

th

e su

rfac

e ar

ea o

f a

sph

ere

as

th

e to

tal

area

of

all

of t

he

non

over

lapp

ing

stri

ps i

t w

ould

tak

e to

cov

er

the

sph

ere.

If

r is

th

e ra

diu

s of

th

e sp

her

e, t

hen

th

e ar

ea o

f a

grea

t ci

rcle

of

the

sph

ere

is π

r2 . T

he

tota

l su

rfac

e ar

ea o

f th

e sp

her

e is

fou

r ti

mes

th

e ar

ea

of a

gre

at c

ircl

e.

F

ind

th

e su

rfac

e ar

ea o

f a

sph

ere

to t

he

nea

rest

ten

th

if t

he

rad

ius

of t

he

sph

ere

is 6

cen

tim

eter

s.

S =

4π

r2 S

urf

ace a

rea o

f a s

phere

=

4π

(6)2

r =

6

≈

452

.4

Sim

plif

y.

Th

e su

rfac

e ar

ea i

s 45

2.4

squ

are

cen

tim

eter

s.

Exer

cise

sF

ind

th

e su

rfac

e ar

ea o

f ea

ch s

ph

ere

or h

emis

ph

ere.

Rou

nd

to

the

nea

rest

ten

th.

1.

5 m

314.2

m2

2.

7 in

153.9

in

2

3.

3 ft

84.8

ft2

4.

9 cm

190.9

cm

2

5. s

pher

e: c

ircu

mfe

ren

ce o

f gr

eat

circ

le =

π c

m 3.1

cm

2

6.

h

em

isp

he

re

: a

re

a o

f g

re

at c

irc

le

≈ 4

π f

t2 37.7

ft2

r

6 cm

Su

rfa

ce

Are

a

of

a S

ph

ere

If a

sp

he

re h

as a

su

rfa

ce

are

a o

f S

sq

ua

re u

nits a

nd

a r

ad

ius o

f r

un

its,

the

n S

= 4

πr2

.

Exam

ple

12-6

026_

044_

GE

OC

RM

C12

_890

521.

indd

375/

30/0

93:

09:0

1P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

3

8

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Su

rface A

reas a

nd

Vo

lum

es o

f S

ph

ere

s

12-6

Vo

lum

es o

f Sp

her

es A

sph

ere

has

on

e ba

sic

mea

sure

men

t, t

he

le

ngt

h o

f it

s ra

diu

s. I

f yo

u k

now

th

e le

ngt

h o

f th

e ra

diu

s of

a s

pher

e, y

ou

can

cal

cula

te i

ts v

olu

me.

F

ind

th

e vo

lum

e of

a s

ph

ere

wit

h r

adiu

s 8

cen

tim

eter

s.

V =

4 −

3 π

r3 V

olu

me o

f a s

phere

=

4 −

3 π

(8)3

r =

8

≈

214

4.7

Sim

plif

y.

Th

e vo

lum

e is

abo

ut

2144

.7 c

ubi

c ce

nti

met

ers.

Exer

cise

sF

ind

th

e vo

lum

e of

eac

h s

ph

ere

or h

emis

ph

ere.

Rou

nd

to

the

nea

rest

ten

th.

1.

5 ft

2.

6

in.

3.

16 in

.

5

23.6

ft3

4

52.4

in

3

8578.6

in

3

4. h

emis

pher

e: r

adiu

s 5

in.

261.8

in

3

5. s

pher

e: c

ircu

mfe

ren

ce o

f gr

eat

circ

le ≈

25

ft 2

63.9

ft3

6. h

emis

pher

e: a

rea

of g

reat

cir

cle

≈ 5

0 m

2 133.0

m3

r

8 cm

Vo

lum

e o

f

a S

ph

ere

If a

sp

he

re h

as a

vo

lum

e o

f V

cu

bic

un

its a

nd

a r

ad

ius o

f r

un

its,

the

n V

= 4

−

3 π

r3.

Exam

ple

026_

044_

GE

OC

RM

C12

_890

521.

indd

383/

27/1

010

:47:

18P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-6

Cha

pte

r 12

39

Gle

ncoe

Geo

met

ry

Skill

s Pr

acti

ceS

urf

ace A

reas a

nd

Vo

lum

es o

f S

ph

ere

sF

ind

th

e su

rfac

e ar

ea o

f ea

ch s

ph

ere

or h

emis

ph

ere.

Rou

nd

to

the

nea

rest

ten

th.

1.

7 in

.

2.

32 m

6

15.8

in

2

3217.0

m2

3. h

emis

pher

e: r

adiu

s of

gre

at c

ircl

e =

8 y

d 603.2

yd

2

4. s

pher

e: a

rea

of g

reat

cir

cle

≈ 2

8.6

in2

114.4

in

2

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e. R

oun

d t

o th

e n

eare

st t

enth

.

5.

16.2

cm

6.

94.8

ft

2

226.1

cm

3

446,0

91.2

ft3

7. h

emis

pher

e: d

iam

eter

= 4

8 yd

28,9

52.9

yd

3

8. s

pher

e: c

ircu

mfe

ren

ce o

f a

grea

t ci

rcle

≈ 2

6 m

296.8

m3

9. s

pher

e: d

iam

eter

= 1

0 in

. 523.6

in

3

12-6

026_

044_

GE

OC

RM

C12

_890

521.

indd

395/

30/0

93:

09:1

6P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

4

0

Gle

ncoe

Geo

met

ry

12-6

Prac

tice

Su

rface A

reas a

nd

Vo

lum

es o

f S

ph

ere

sF

ind

th

e su

rfac

e ar

ea o

f ea

ch s

ph

ere

or h

emis

ph

ere.

Rou

nd

to

the

nea

rest

ten

th.

1.

6.5

cm

2.

89 ft

5

30.9

cm

2

24,8

84.6

ft2

3. h

emis

pher

e: r

adiu

s of

gre

at c

ircl

e =

8.4

in

. 665.0

in

2

4. s

pher

e: a

rea

of g

reat

cir

cle

≈ 2

9.8

m2

119.2

m2

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e. R

oun

d t

o th

e n

eare

st t

enth

.

5.

12.3

2 ft

6.

32

m

7

832.9

ft3

8578.6

m3

7. h

emis

pher

e: d

iam

eter

= 1

8 m

m 1

526.8

mm

3

8. s

pher

e: c

ircu

mfe

ren

ce ≈

36

yd 7

87.9

yd

3

9. s

pher

e: r

adiu

s =

12.

4 in

. 7986.4

in

3

026_

044_

GE

OC

RM

C12

_890

521.

indd

405/

30/0

93:

09:2

1P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-6

Cha

pte

r 12

41

Gle

ncoe

Geo

met

ry

Wor

d Pr

oble

m P

ract

ice

Su

rface A

reas a

nd

Vo

lum

es o

f S

ph

ere

s 1

. OR

AN

GES

Man

dy c

uts

a s

pher

ical

or

ange

in

hal

f al

ong

a gr

eat

circ

le. I

f th

e ra

diu

s of

th

e or

ange

is

2 in

ches

, wh

at i

s th

e ar

ea o

f th

e cr

oss

sect

ion

th

at M

andy

cu

t? R

oun

d yo

ur

answ

er t

o th

e n

eare

st

hu

ndr

edth

.

1

2.5

7 i

n2

2. B

ILLI

AR

DS

A b

illi

ard

ball

set

con

sist

s of

16

sph

eres

, eac

h 2

1 − 4 i

nch

es i

n

diam

eter

. Wh

at i

s th

e to

tal

volu

me

of

a co

mpl

ete

set

of b

illi

ard

ball

s? R

oun

d yo

ur

answ

er t

o th

e n

eare

st t

hou

san

dth

of

a c

ubi

c in

ch.

95.4

26 i

n3

3. M

OO

NS

OF

SATU

RN

Th

e pl

anet

S

atu

rn h

as s

ever

al m

oon

s. T

hes

e ca

n

be m

odel

ed a

ccu

rate

ly b

y sp

her

es.

Sat

urn

’s l

arge

st m

oon

Tit

an h

as a

ra

diu

s of

abo

ut

2575

kil

omet

ers.

Wh

at i

s th

e ap

prox

imat

e su

rfac

e ar

ea o

f T

itan

? R

oun

d yo

ur

answ

er t

o th

e n

eare

st t

enth

.

83,3

22,8

91.2

km

2

4. T

HE

ATM

OSP

HER

EA

bou

t 99

% o

f

Ear

th’s

atm

osph

ere

is c

onta

ined

in

a

31-k

ilom

eter

th

ick

laye

r th

at e

nw

raps

th

e pl

anet

. Th

e E

arth

its

elf

is a

lmos

t a

sph

ere

wit

h r

adiu

s 63

78 k

ilom

eter

s.

Wh

at i

s th

e ra

tio

of t

he

volu

me

of t

he

atm

osph

ere

to t

he

volu

me

of E

arth

? R

oun

d yo

ur

answ

er t

o th

e n

eare

st

thou

san

dth

.

0.0

15

5. C

UB

ESM

arcu

s bu

ilds

a s

pher

e in

side

of

a cu

be. T

he

sph

ere

fits

sn

ugl

y in

side

th

e cu

be s

o th

at t

he

sph

ere

tou

ches

th

e cu

be

at o

ne

poin

t on

eac

h s

ide.

Th

e si

de

len

gth

of

the

cube

is

2 in

ches

.

a. W

hat

is

the

surf

ace

area

of

the

cube

?

24 i

n2

b.

Wh

at i

s th

e su

rfac

e ar

ea o

f th

e sp

her

e? R

oun

d yo

ur

answ

ers

to t

he

nea

rest

hu

ndr

edth

.

12.5

7 i

n2

c. W

hat

is

the

rati

o of

th

e su

rfac

e ar

ea

of t

he

cube

to

the

surf

ace

area

of

the

sph

ere?

Rou

nd

you

r an

swer

to

the

nea

rest

hu

ndr

edth

.

1.9

1

12-6

026_

044_

GE

OC

RM

C12

_890

521.

indd

415/

30/0

93:

09:2

6P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

4

2

Gle

ncoe

Geo

met

ry

Enri

chm

ent

Sp

here

s a

nd

Den

sit

yT

he

den

sity

of

a m

etal

is

a ra

tio

of i

ts m

ass

to i

ts v

olu

me.

For

ex

ampl

e, t

he

mas

s of

alu

min

um

is

2.7

gram

s pe

r cu

bic

cen

tim

eter

. H

ere

is a

lis

t of

sev

eral

met

als

and

thei

r de

nsi

ties

.

Alu

min

um

2.

7 g/

cm3

Cop

per

8.96

g/c

m3

Gol

d 19

.32

g/cm

3 Ir

on

7.87

4 g/

cm3

Lea

d 11

.35

g/cm

3 P

lati

nu

m

21.4

5 g/

cm3

Sil

ver

10.5

0 g/

cm3

To

calc

ula

te t

he

mas

s of

a p

iece

of

met

al, m

ult

iply

vol

um

e by

den

sity

.

F

ind

th

e m

ass

of a

sil

ver

bal

l th

at i

s 0.

8 cm

in

dia

met

er.

M =

D ·

V

= 1

0.5

· 4 −

3 π

(0.4

)3

≈ 1

0.5(

0.27

)≈

2.8

1

Th

e m

ass

is a

bou

t 2.

81 g

ram

s.

Exer

cise

sF

ind

th

e m

ass

of e

ach

met

al b

all

des

crib

ed. A

ssu

me

the

bal

ls a

re

sph

eric

al. R

oun

d y

our

answ

ers

to t

he

nea

rest

ten

th.

1. a

cop

per

ball

1.2

cm

in

dia

met

er 8

.1 g

2. a

gol

d ba

ll 0

.6 c

m i

n d

iam

eter

2.2

g

3. a

n a

lum

inu

m b

all

wit

h r

adiu

s 3

cm 3

05.4

g

4. a

pla

tin

um

bal

l w

ith

rad

ius

0.7

cm 3

0.8

g

Sol

ve. A

ssu

me

the

bal

ls a

re s

ph

eric

al. R

oun

d y

our

answ

ers

to t

he

nea

rest

ten

th.

5. A

lea

d ba

ll w

eigh

s 32

6 gr

ams.

Fin

d th

e ra

diu

s of

th

e ba

ll t

o th

e n

eare

st

ten

th o

f a

cen

tim

eter

. 1.9

cm

6. A

n i

ron

bal

l w

eigh

s 80

4 gr

ams.

Fin

d th

e di

amet

er o

f th

e ba

ll t

o th

e n

eare

st t

enth

of

a ce

nti

met

er.

5.8

cm

7. A

sil

ver

ball

an

d a

copp

er b

all

each

hav

e a

diam

eter

of

3.5

cen

tim

eter

s.

Wh

ich

wei

ghs

mor

e? H

ow m

uch

mor

e? s

ilver;

34.6

g

8. A

n a

lum

inu

m b

all

and

a le

ad b

all

each

hav

e a

radi

us

of 1

.2 c

enti

met

ers.

W

hic

h w

eigh

s m

ore?

How

mu

ch m

ore?

lead

; 62.6

g

12-6

Exam

ple

026_

044_

GE

OC

RM

C12

_890

521.

indd

425/

30/0

93:

09:3

1P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 12-7

Cha

pte

r 12

43

Gle

ncoe

Geo

met

ry

12-7

Stud

y G

uide

and

Inte

rven

tion

Sp

heri

cal

Geo

metr

y

Geo

met

ry O

n A

Sp

her

e U

p to

now

, we

hav

e be

en s

tudy

ing

Eu

clid

ean

geo

met

ry,

wh

ere

a pl

ane

is a

fla

t su

rfac

e m

ade

up

of p

oin

ts t

hat

ext

ends

in

fin

itel

y in

all

dir

ecti

ons.

In

sp

her

ical

geo

met

ry, a

pla

ne

is t

he

surf

ace

of a

sph

ere.

Nam

e ea

ch o

f th

e fo

llow

ing

on s

ph

ere

K.

a. t

wo

lin

es c

onta

inin

g th

e p

oin

t F

� ��

EG

an

d � �

��

BH

are

lin

es o

n s

pher

e K

th

at c

onta

in t

he

poin

t F

b. a

lin

e se

gmen

t co

nta

inin

g th

e p

oin

t J

−

−

ID

is

a se

gmen

t on

sph

ere

K t

hat

con

tain

s th

e po

int

J

c. a

tri

angl

e

�

AH

I is

a t

rian

gle

on s

pher

e K

Exer

cise

sN

ame

two

lin

es c

onta

inin

g p

oin

t Z

, a s

egm

ent

con

tain

ing

poi

nt

R, a

nd

a t

rian

gle

in e

ach

of

the

foll

owin

g sp

her

es.

1.

F

2.

M

�

⎯

�

WT a

nd

�

⎯

�

SV

, −

−

X

U a

nd

�S

RT

�

⎯

�

AD

an

d �

⎯

�

EB

, −

−

G

C a

nd

�G

RZ

Det

erm

ine

wh

eth

er f

igu

re u

on

eac

h o

f th

e sp

her

es s

how

n i

s a

lin

e in

sp

her

ical

ge

omet

ry.

3.

No

4.

N

o

5. G

EOG

RA

PHY

Lin

es o

f la

titu

de r

un

hor

izon

tall

y ac

ross

th

e su

rfac

e of

Ear

th. A

re t

her

e an

y li

nes

of

lati

tude

th

at a

re g

reat

cir

cles

? E

xpla

in.

Y

es,

the e

qu

ato

r is

a g

reat

cir

cle

. N

o o

ther

lin

e o

f la

titu

de i

s a

gre

at

cir

cle

.

Exam

ple

K

026_

044_

GE

OC

RM

C12

_890

521.

indd

435/

30/0

93:

09:3

6P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

4

4

Gle

ncoe

Geo

met

ry

Co

mp

arin

g E

ucl

idea

n a

nd

Sp

her

ical

Geo

met

ries

Som

e po

stu

late

s an

d pr

oper

ties

of

Eu

clid

ean

geo

met

ry a

re t

rue

in s

pher

ical

geo

met

ry. O

ther

s ar

e n

ot t

rue

or a

re

tru

e on

ly u

nde

r ce

rtai

n c

ircu

mst

ance

s.

Tel

l w

het

her

th

e fo

llow

ing

pos

tula

te o

r p

rop

erty

of

pla

ne

Eu

clid

ean

ge

omet

ry h

as a

cor

resp

ond

ing

stat

emen

t in

sp

her

ical

geo

met

ry. I

f so

, wri

te t

he

corr

esp

ond

ing

stat

emen

t. I

f n

ot, e

xpla

in y

our

reas

onin

g.

Giv

en a

ny

lin

e, t

her

e ar

e an

in

fin

ite

nu

mb

er o

f p

aral

lel

lin

es.

On

th

e sp

her

e to

th

e ri

ght,

if

we

are

give

n l

ine

m w

e se

e th

at i

t go

es t

hro

ugh

th

e po

les

of t

he

sph

ere.

If

we

try

to m

ake

any

oth

er

lin

e on

th

e sp

her

e, i

t w

ould

in

ters

ect

lin

e m

at

exac

tly

2 po

ints

. T

his

pro

pert

y is

not

tru

e in

sph

eric

al g

eom

etry

.A

cor

resp

ondi

ng

stat

emen

t in

sph

eric

al g

eom

etry

wou

ld b

e:

“Giv

en a

ny

lin

e, t

her

e ar

e n

o pa

rall

el l

ines

.”

Exer

cise

sT

ell

wh

eth

er t

he

foll

owin

g p

ostu

late

or

pro

per

ty o

f p

lan

e E

ucl

idea

n

geom

etry

has

a c

orre

spon

din

g st

atem

ent

in s

ph

eric

al g

eom

etry

. If

so,

wri

te t

he

corr

esp

ond

ing

stat

emen

t. I

f n

ot, e

xpla

in y

our

reas

onin

g.

1. I

f tw

o n

onid

enti

cal

lin

es i

nte

rsec

t at

a p

oin

t, t

hey

do

not

in

ters

ect

agai

n.

N

o.

If t

wo

no

nid

en

tical

lin

es i

nte

rsect

at

a p

oin

t, t

hey i

nte

rsect

ag

ain

o

n t

he o

pp

osit

e s

ide o

f th

e s

ph

ere

.

2. G

iven

a l

ine

and

a po

int

on t

he

lin

e, t

her

e is

on

ly o

ne

perp

endi

cula

r li

ne

goin

g th

rou

gh

that

poi

nt.

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

3. G

iven

tw

o pa

rall

el l

ines

an

d a

tran

sver

sal,

alte

rnat

e in

teri

or a

ngl

es a

re c

ongr

uen

t.

No

. T

here

are

no

para

llel

lin

es i

n s

ph

eri

cal

geo

metr

y.

4. I

f tw

o li

nes

are

per

pen

dicu

lar

to a

th

ird

lin

e, t

hey

are

par

alle

l.

No

. T

here

are

no

para

llel

lin

es i

n s

ph

eri

cal

geo

metr

y.

5. T

hre

e n

onco

llin

ear

poin

ts d

eter

min

e a

tria

ngl

e.

Yes.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

6. A

lar

gest

an

gle

of a

tri

angl

e is

opp

osit

e th

e la

rges

t si

de.

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

12-7

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Sp

heri

cal

Geo

metr

y

Exam

ple

026_

044_

GE

OC

RM

C12

_890

521.

indd

444/

10/0

89:

12:3

3P

M

Lesson 12-7


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

45

Gle

ncoe

Geo

met

ry

Nam

e tw

o li

nes

con

tain

ing

poi

nt

K, a

seg

men

t co

nta

inin

g p

oin

t T

, an

d a

tri

angl

e in

eac

h o

f th

e fo

llow

ing

sph

eres

.

1.

C

2.

L

Det

erm

ine

wh

eth

er f

igu

re u

on

eac

h o

f th

e sp

her

es s

how

n i

s a

lin

e in

sp

her

ical

geo

met

ry.

3.

4.

bask

etba

ll

Tel

l w

het

her

th

e fo

llow

ing

pos

tula

te o

r p

rop

erty

of

pla

ne

Eu

clid

ean

geo

met

ry h

as

a co

rres

pon

din

g st

atem

ent

in s

ph

eric

al g

eom

etry

. If

so, w

rite

th

e co

rres

pon

din

g st

atem

ent.

If

not

, exp

lain

you

r re

ason

ing.

5. I

f tw

o li

nes

for

m v

erti

cal

angl

es, t

hen

th

e an

gles

are

equ

al i

n m

easu

re.

6. I

f tw

o li

nes

mee

t a

thir

d li

ne

at t

he

sam

e an

gle,

th

ose

lin

es a

re p

aral

lel.

7. T

wo

lin

es m

eet

at t

wo

90°

angl

es o

r th

ey m

eet

at a

ngl

es w

hos

e su

m i

s 18

0°.

8. T

hre

e n

on-p

aral

lel

lin

es d

ivid

e th

e pl

ane

into

7 s

epar

ate

part

s.

Skill

s Pr

acti

ceS

ph

eri

cal

Geo

metr

y

12-7

�

⎯

�

SF a

nd

�

⎯

�

IH , −

−

S

G ,

an

d �

STH

�

⎯

�

BD

an

d �

⎯

�

ET ,

−−

C

A ,

an

d �

ATE

N

o

Yes

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

N

o.

Th

ere

are

no

para

llel

lin

es i

n s

ph

eri

cal

geo

metr

y.

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

N

o.

Th

ree l

ines d

ivid

e t

he p

lan

e i

nto

6 o

r 7 s

ep

ara

te p

art

s.

045_

055_

GE

OC

RM

C12

_890

521.

indd

456/

2/09

4:48

:36

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

4

6

Gle

ncoe

Geo

met

ry

Nam

e tw

o li

nes

con

tain

ing

poi

nt

K, a

seg

men

t co

nta

inin

g p

oin

t T

, an

d a

tri

angl

e in

eac

h o

f th

e fo

llow

ing

sph

eres

.

1.

L

2.

M

Det

erm

ine

wh

eth

er f

igu

re u

on

eac

h o

f th

e sp

her

es s

how

n i

s a

lin

e in

sp

her

ical

ge

omet

ry.

3.

tenn

is b

all

4.

Tel

l w

het

her

th

e fo

llow

ing

pos

tula

te o

r p

rop

erty

of

pla

ne

Eu

clid

ean

geo

met

ry h

as

a co

rres

pon

din

g st

atem

ent

in s

ph

eric

al g

eom

etry

. If

so, w

rite

th

e co

rres

pon

din

g st

atem

ent.

If

not

, exp

lain

you

r re

ason

ing.

5. A

tri

angl

e ca

n h

ave

at m

ost

one

obtu

se a

ngl

e.

N

o.

A t

rian

gle

can

have a

t m

ost

thre

e o

btu

se a

ng

les.

6. T

he

sum

of

the

angl

es o

f a

tria

ngl

e is

180

°.

N

o.

Th

e s

um

of

the a

ng

les o

f a t

rian

gle

is m

ore

th

an

180°.

7. G

iven

a l

ine

and

a po

int

not

on

th

e li

ne,

th

ere

is e

xact

ly o

ne

lin

e th

at g

oes

thro

ugh

th

e po

int

and

is p

erpe

ndi

cula

r to

th

e li

ne.

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

8. A

ll e

quil

ater

al t

rian

gles

are

sim

ilar

.

Y

es.

Th

e s

am

e s

tate

men

t w

ork

s i

n s

ph

eri

cal

geo

metr

y.

9. A

IRPL

AN

ES W

hen

fly

ing

an a

irpl

ane

from

New

Yor

k to

Sea

ttle

, wh

at i

s th

e sh

orte

st

rou

te: f

lyin

g di

rect

ly w

est,

or

flyi

ng

nor

th a

cros

s C

anad

a? E

xpla

in.

T

he s

ho

rtest

rou

te i

s t

o f

ly a

lon

g t

he g

reat

cir

cle

co

nn

ecti

ng

New

Yo

rk

an

d S

eatt

le,

wh

ich

cro

sses C

an

ad

a.

12-7

Prac

tice

Sp

heri

cal

Geo

metr

y

�

⎯

�

CB

an

d �

⎯

�

AE

, �

⎯

�

FA

, �

AD

E

�

⎯

�

ZX

an

d �

⎯

�

TY

, �

⎯

�

ZX

an

d �

TZ

K

N

o

Yes

045_

055_

GE

OC

RM

C12

_890

521.

indd

466/

2/09

4:46

:23

PM


NA

ME

DA

TE

PE

RIO

D

Lesson 12-7

Cha

pte

r 12

47

Gle

ncoe

Geo

met

ry

12-7

Wor

d Pr

oble

m P

ract

ice

Sp

heri

cal

Geo

metr

y

1. P

AIN

TIN

GC

onsi

der

pain

tin

g qu

adri

late

ral

AB

CD

on

th

e be

ach

bal

l w

ith

rad

ius

1 ft

. Wh

at i

s th

e su

rfac

e ar

ea y

ou w

ould

nee

d to

pai

nt?

2. E

AR

THT

he

Equ

ator

an

d th

e P

rim

e M

erid

ian

are

per

pen

dicu

lar

grea

t ci

rcle

s th

at d

ivid

e E

arth

in

to N

orth

, Sou

th a

nd

Eas

t, W

est

hem

isph

eres

. If

Ear

th h

as a

su

rfac

e ar

ea o

f 19

7,00

0,00

0 sq

uar

e m

iles

, wh

at i

s th

e su

rfac

e ar

ea o

f th

e N

orth

-Eas

t se

ctio

n o

f E

arth

?

Sou

rce:

NA

SA

3. O

CEA

NIf

th

e oc

ean

s co

ver

70%

of

Ear

th’s

su

rfac

e, w

hat

is

the

surf

ace

area

of

th

e oc

ean

s?

Sou

rce:

NA

SA

4. G

EOM

ETR

YT

hre

e n

onid

enti

cal

lin

es o

n

the

circ

le d

ivid

e it

in

to e

ith

er

6 se

ctio

ns

or 8

tri

angl

es. W

hat

con

diti

on

is n

eede

d so

th

at t

he

thre

e li

nes

for

m

6 se

ctio

ns?

5. G

EOG

RA

PHY

Lat

itu

de a

nd

lon

gitu

de

lin

es a

re i

mag

inar

y li

nes

on

Ear

th. T

he

lin

es o

f la

titu

de a

re h

oriz

onta

l co

nce

ntr

ic

circ

les

that

hel

p to

def

ine

the

dist

ance

a

plac

e is

fro

m t

he

equ

ator

. Lin

es o

f la

titu

de a

re m

easu

red

in d

egre

es. T

he

equ

ator

is

0°. T

he

nor

th p

ole

is 9

0° n

orth

la

titu

de. T

he

lin

es o

f lo

ngi

tude

are

gre

at

circ

les

that

hel

p to

def

ine

the

dist

ance

a

plac

e is

fro

m t

he

Pri

me

Mer

idan

, wh

ich

is

loc

ated

in

En

glan

d an

d co

nsi

dere

d th

e lo

ngi

tude

of

0°.

a.

Th

e m

ean

rad

ius

of E

arth

is

3963

m

iles

. Atl

anta

, Geo

rgia

, has

co

ordi

nat

es (

33°N

, 84°

W)

and

Cin

cin

nat

i, O

hio

, has

coo

rdin

ates

(3

9°N

, 84°

W).

Est

imat

e th

e di

stan

ce

betw

een

Atl

anta

an

d C

inci

nn

ati

to

the

nea

rest

ten

th.

b

. S

eatt

le, W

ash

ingt

on, h

as c

oord

inat

es

(47°

N, 1

22°W

) an

d P

ortl

and,

Ore

gon

,h

as c

oord

inat

es (

45°N

, 122

°W).

E

stim

ate

the

dist

ance

bet

wee

n

Por

tlan

d an

d S

eatt

le t

o th

e n

eare

st

ten

th.

A C

DB

6π

−

7

sq

uare

feet

2

4,6

25,0

00 s

qu

are

miles

1

37,9

00,0

00 s

qu

are

miles

A

ll t

hre

e l

ines m

ust

inte

rsect

at

two

po

ints

.

4

15.4

miles

138.3

miles

045_

055_

GE

OC

RM

C12

_890

521.

indd

475/

30/0

93:

10:1

8P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

4

8

Gle

ncoe

Geo

met

ry

12-7

Pro

jecti

on

sW

hen

mak

ing

map

s of

Ear

th, c

arto

grap

her

s m

ust

sh

ow a

sph

ere

on a

pla

ne.

To

do t

his

th

ey

hav

e to

use

pro

ject

ion

s, a

met

hod

of

con

vert

ing

a sp

her

e in

to a

pla

ne.

Bu

t th

ese

proj

ecti

ons

hav

e th

eir

lim

itat

ion

s.

Th

e m

ap o

n t

he

righ

t is

a M

erca

tor

proj

ecti

on

of E

arth

. On

th

is m

ap G

reen

lan

d ap

pear

s to

be

the

sam

e si

ze a

s A

fric

a. B

ut

Gre

enla

nd

has

a l

and

area

of

2,16

6,08

6 sq

uar

e ki

lom

eter

s an

d A

fric

a h

as a

lan

d ar

ea o

f 30

,365

,700

squ

are

kilo

met

ers.

Th

e m

ap o

n t

he

righ

t is

a L

ambe

rt p

roje

ctio

n.

Wh

en a

pil

ot d

raw

s a

stra

igh

t li

ne

betw

een

tw

o po

ints

on

th

is m

ap t

he

lin

e sh

ows

tru

e be

arin

g, o

r re

lati

ve d

irec

tion

to

the

Nor

th P

ole.

How

ever

, th

e bo

ttom

are

a of

th

is m

ap d

isto

rts

dist

ance

s.

1. W

hen

wou

ld i

t be

use

ful

to u

se a

Mer

cato

r pr

ojec

tion

of

Ear

th?

T

o m

easu

re d

ista

nces c

lose t

o t

he e

qu

ato

r.

2. D

oes

each

squ

are

on t

he

Mer

cato

r pr

ojec

tion

hav

e th

e sa

me

surf

ace

area

? E

xpla

in.

N

o.

Near

the t

op

, th

e w

idth

acro

ss i

s m

uch

sm

aller

than

th

e e

qu

ato

r.

3. D

oes

each

squ

are

on t

he

Lam

bert

pro

ject

ion

hav

e th

e sa

me

surf

ace

area

? E

xpla

in.

N

o.

Lin

es o

f lo

ng

itu

de b

eco

me s

ho

rter

as y

ou

tra

vel

tow

ard

s t

he p

ole

s.

4. T

he

Mer

cato

r pr

ojec

tion

use

s a

cyli

nde

r to

map

Ear

th, w

hil

e th

e L

ambe

rt p

roje

ctio

n u

ses

a co

ne

to m

ap E

arth

. Wh

at o

ther

sh

apes

do

you

th

ink

cou

ld b

e u

sed

to m

ap E

arth

?

T

rian

gle

s,

hexag

on

s,

pri

sm

s.

Enri

chm

ent

Sp

heri

cal

Geo

metr

y

60˚ N

180˚

W18

0˚ E

90˚ W

0˚90

˚ E

40˚ N

20˚ N

0˚ 20˚ S

40˚ S

60˚ S

045_

055_

GE

OC

RM

C12

_890

521.

indd

484/

12/0

89:

02:1

0P

M

Lesson 12-8


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

49

Gle

ncoe

Geo

met

ry

Iden

tify

Co

ng

ruen

t o

r Si

mila

r So

lids

Sim

ilar

sol

ids

hav

e ex

actl

y th

e sa

me

shap

e bu

t n

ot n

eces

sari

ly t

he

sam

e si

ze. T

wo

soli

ds a

re s

imil

ar i

f th

ey a

re t

he

sam

e sh

ape

and

the

rati

os o

f th

eir

corr

espo

ndi

ng

lin

ear

mea

sure

s ar

e eq

ual

. All

sph

eres

are

sim

ilar

an

d al

l cu

bes

are

sim

ilar

. Con

gru

ent

soli

ds

hav

e ex

actl

y th

e sa

me

shap

e an

d th

e sa

me

size

. C

ongr

uen

t so

lids

are

sim

ilar

sol

ids

wit

h a

sca

le f

acto

r of

1:1

. Con

gru

ent

soli

ds h

ave

the

foll

owin

g ch

arac

teri

stic

s:

• C

orre

spon

din

g an

gles

are

con

gru

ent

• C

orre

spon

din

g ed

ges

are

con

gru

ent

• C

orre

spon

din

g fa

ces

are

con

gru

ent

• V

olu

mes

are

equ

al

D

eter

min

e w

het

her

th

e p

air

of s

olid

s is

sim

ila

r, c

ongr

uen

t, o

r n

eith

er. I

f th

e so

lid

s ar

e si

mil

ar, s

tate

th

e sc

ale

fact

or.

rati

o of

wid

th:

3 −

6 =

1 −

2

rati

o of

len

gth

: 4 −

8 =

1 −

2

rati

o of

hyp

oten

use

: 5 −

10

= 1 −

2

rati

o of

hei

ght:

4 −

8 =

1 −

2

Th

e ra

tios

of

the

corr

espo

ndi

ng

side

s ar

e eq

ual

, so

the

tria

ngu

lar

pris

ms

are

sim

ilar

. Th

e sc

ale

fact

or i

s 1:

2. S

ince

th

e sc

ale

fact

or i

s n

ot 1

:1, t

he

soli

ds a

re n

ot c

ongr

uen

t.

Exer

cise

sD

eter

min

e w

het

her

th

e p

air

of s

olid

s is

sim

ila

r, c

ongr

uen

t, o

r n

eith

er. I

f th

e so

lid

s ar

e si

mil

ar, s

tate

th

e sc

ale

fact

or.

1.

2 cm

1 cm

10 c

m

5 cm

2.

4.2

in.

12.3

in.

4.2

in.

12.3

in.

3.

4 in

.

8 in

.

4.

2 m

2 m

4 m

1 m

1 m

3 m

Stud

y G

uide

and

Inte

rven

tion

Co

ng

ruen

t an

d S

imilar

So

lid

s

12-8

5 in

.

4 in

.

4 in

.3

in.

10 in

.

8 in

.

8 in

.

6 in

.Ex

amp

le

S

imilar,

1:5

C

on

gru

en

t

S

imilar,

1:2

N

eit

her

045_

055_

GE

OC

RM

C12

_890

521.

indd

495/

30/0

93:

10:3

3P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

5

0

Gle

ncoe

Geo

met

ry

Pro

per

ties

of

Co

ng

ruen

t o

r Si

mila

r So

lids

Wh

en p

airs

of

soli

ds a

re c

ongr

uen

t or

si

mil

ar, c

erta

in p

rope

rtie

s ar

e kn

own

.

If t

wo

sim

ilar

sol

ids

hav

e a

scal

e fa

ctor

of

a:b

then

,•

the

rati

o of

th

eir

surf

ace

area

s is

a2 :b

2 .•

the

rati

o of

th

eir

volu

mes

is

a3 :b3 .

T

wo

sph

eres

hav

e ra

dii

of

2 fe

et a

nd

6 f

eet.

W

hat

is

the

rati

o of

th

e vo

lum

e of

th

e sm

all

sph

ere

to t

he

volu

me

of t

he

larg

e sp

her

e?

Fir

st, f

ind

the

scal

e fa

ctor

.

radi

us

of t

he

smal

l sp

her

e

−

−

rad

ius

of t

he

larg

e sp

her

e = 2 −

6 o

r 1 −

3

Th

e sc

ale

fact

or i

s 1 −

3 .

a3 −

b3

= (1

)3 −

(3

)3 or

1 −

27

So,

th

e ra

tio

of t

he

volu

mes

is

1:27

.

Exer

cise

s 1

. Tw

o cu

bes

hav

e si

de l

engt

hs

of 3

in

ches

an

d 8

inch

es. W

hat

is

the

rati

o of

th

e su

rfac

e ar

ea o

f th

e sm

all

cube

to

the

surf

ace

area

of

the

larg

e cu

be?

9

:64

2. T

wo

sim

ilar

con

es h

ave

hei

ghts

of

3 fe

et a

nd

12 f

eet.

Wh

at i

s th

e ra

tio

of t

he

volu

me

of

the

smal

l co

ne

to t

he

volu

me

of t

he

larg

e co

ne?

1

:64

3. T

wo

sim

ilar

tri

angu

lar

pris

ms

hav

e vo

lum

es o

f 27

squ

are

met

ers

and

64 s

quar

e m

eter

s.

Wh

at i

s th

e ra

tio

of t

he

surf

ace

area

of

the

smal

l pr

ism

to

the

surf

ace

area

of

the

larg

e pr

ism

?

9

:16

4. C

OM

PUTE

RS

A s

mal

l re

ctan

gula

r la

ptop

has

a w

idth

of

10 i

nch

es a

nd

an a

rea

of

80 s

quar

e in

ches

. A l

arge

r an

d si

mil

ar l

apto

p h

as a

wid

th o

f 15

in

ches

. Wh

at i

s th

e le

ngt

h o

f th

e la

rger

lap

top?

1

2 i

n.

5. C

ON

STR

UC

TIO

N A

bu

ildi

ng

com

pan

y u

ses

two

sim

ilar

siz

es o

f pi

pes.

Th

e sm

alle

r si

ze

has

a r

adiu

s of

1 i

nch

an

d le

ngt

h o

f 8

inch

es. T

he

larg

er s

ize

has

a r

adiu

s of

2.5

in

ches

W

hat

is

the

volu

me

of t

he

larg

er p

ipes

?

1

25

π c

ub

ic i

n.

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Co

ng

ruen

t an

d S

imilar

So

lid

s

Exam

ple

12-8

2 ft

6 ft

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NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

51

Gle

ncoe

Geo

met

ry

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

is s

imil

ar,

con

gru

ent,

or

nei

ther

. If

the

soli

ds

are

sim

ilar

, sta

te t

he

scal

e fa

ctor

.

1.

2 cm

4 cm3

cm

6 cm

12 c

m

9 cm

2.

6 cm

9 cm

8 cm

12 c

m

3.

5 m

10 m

4.

3 ft

1 ft

1 ft

9 ft

3 ft

3 ft

5. T

wo

sim

ilar

pyr

amid

s h

ave

hei

ghts

of

4 in

ches

an

d 7

inch

es W

hat

is

the

rati

o of

th

e vo

lum

e of

th

e sm

all

pyra

mid

to

the

volu

me

of t

he

larg

e py

ram

id?

6

4:3

43

6. T

wo

sim

ilar

cyl

inde

rs h

ave

surf

ace

area

s of

40π

squ

are

feet

an

d 90

π s

quar

e fe

et. W

hat

is

th

e ra

tio

of t

he

hei

ght

of t

he

larg

e cy

lin

der

to t

he

hei

ght

of t

he

smal

l cy

lin

der?

3

:2

7. C

OO

KIN

G T

wo

stoc

kpot

s ar

e si

mil

ar c

ylin

ders

. Th

e sm

alle

r st

ockp

ot h

as a

hei

ght

of

10 i

nch

es a

nd

a ra

diu

s of

2.5

in

ches

. Th

e la

rger

sto

ckpo

t h

as a

hei

ght

of 1

6 in

ches

. Wh

at

is t

he

volu

me

of t

he

larg

er s

tock

pot?

Rou

nd

to t

he

nea

rest

ten

th.

Skill

s Pr

acti

ceC

on

gru

en

t an

d S

imilar

So

lid

s

12-8

s

imilar;

1:3

s

imilar;

3:4

c

on

gru

en

t s

imilar;

1:3

804.2

in

3

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An

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Co

pyr

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© G

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raw

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The

McG

raw

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Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

5

2

Gle

ncoe

Geo

met

ry

Det

erm

ine

wh

eth

er t

he

pai

r of

sol

ids

is s

imil

ar,

con

gru

ent,

or

nei

ther

. If

the

soli

ds

are

sim

ilar

, sta

te t

he

scal

e fa

ctor

.

1.

6 cm

8 cm

18 c

m

24 c

m

2.

5 cm

12 c

m

10 c

m

24 c

m

3.

1 m

1 m

5 m

5 m

3 m

3 m

4 m

4 m

4.

5 cm

5 cm

2 cm

10 c

m

10 c

m

1.5

cm

5. T

wo

cube

s h

ave

surf

ace

area

s of

72

squ

are

feet

an

d 98

squ

are

feet

. Wh

at i

s th

e ra

tio

of

the

volu

me

of t

he

smal

l cu

be t

o th

e vo

lum

e of

th

e la

rge

cube

?

2

16:3

43

6. T

wo

sim

ilar

ice

cre

am c

ones

are

mad

e of

a h

alf

sph

ere

on t

op a

nd

a co

ne

on b

otto

m.

Th

ey h

ave

radi

i of

1 i

nch

an

d 1.

75 i

nch

es r

espe

ctiv

ely.

Wh

at i

s th

e ra

tio

of t

he

volu

me

of t

he

smal

l ic

e cr

eam

con

e to

th

e vo

lum

e of

th

e la

rge

ice

crea

m c

one?

Rou

nd

to t

he

nea

rest

ten

th.

1

:5.4

7. A

RC

ITH

ECTU

RE

Arc

hit

ects

mak

e sc

ale

mod

els

of b

uil

din

gs t

o pr

esen

t th

eir

idea

s to

cl

ien

ts. I

f an

arc

hit

ect

wan

ts t

o m

ake

a 1:

50 s

cale

mod

el o

f a

4000

squ

are

foot

hou

se,

how

man

y sq

uar

e fe

et w

ill

the

mod

el h

ave?

1

.6 s

qu

are

feet

12-8

Prac

tice

Co

ng

ruen

t an

d S

imilar

So

lid

s

s

imilar,

1:3

n

eit

her

c

on

gru

en

t n

eit

her

045_

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OC

RM

C12

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521.

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525/

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93:

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Lesson 12-8


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

53

Gle

ncoe

Geo

met

ry

12-8

Wor

d Pr

oble

m P

ract

ice

Co

ng

ruen

t an

d S

imilar

So

lid

s

1. C

OO

KIN

GA

cyl

indr

ical

pot

is

4.5

inch

esta

ll a

nd

has

a r

adiu

s of

4 i

nch

es. H

ow

tall

wou

ld a

sim

ilar

pot

be

if i

ts r

adiu

s is

6

inch

es?

2. M

AN

UFA

CTU

RIN

GB

oxes

, In

c. w

ants

to

mak

e th

e tw

o bo

xes

belo

w. H

ow l

ong

does

th

e se

con

d bo

x n

eed

to b

e so

th

at

they

are

sim

ilar

?

24 c

m

15 c

m25

cm

25 c

m15

cm

3. F

AR

MIN

GA

far

mer

has

tw

o si

mil

ar

cyli

ndr

ical

gra

in s

ilos

. Th

e sm

alle

r si

lo

is 2

5 fe

et t

all

and

the

larg

er s

ilo

is

40 f

eet

tall

. If

the

smal

ler

silo

can

hol

d 15

00 c

ubi

c fe

et o

f gr

ain

, how

mu

ch c

an

the

larg

er s

ilo

hol

d?

4. P

LAN

ETS

Ear

th h

as a

su

rfac

e ar

ea o

f ab

out

196,

937,

500

squ

are

mil

es. M

ars

has

a s

urf

ace

area

of

abou

t 89

,500

,000

sq

uar

e m

iles

. Wh

at i

s th

e ra

tio

of t

he

radi

us

of E

arth

to

the

radi

us

of M

ars?

R

oun

d to

th

e n

eare

st t

enth

.

Sou

rce:

NA

SA

5. B

ASE

BA

LLM

ajor

Lea

gue

Bas

ebal

l or

M

LB

, ru

les

stat

e th

at b

aseb

alls

mu

st

hav

e a

circ

um

fere

nce

of

9 in

ches

. Th

e N

atio

nal

Sof

tbal

l A

ssoc

iati

on, o

r N

SA

, ru

les

stat

e th

at s

oftb

alls

mu

st h

ave

a ci

rcu

mfe

ren

ce n

ot e

xcee

din

g 12

in

ches

.

Sou

rce:

ML

B, N

SA

a.

Fin

d th

e ra

tio

of t

he

circ

um

fere

nce

of

ML

B b

aseb

alls

to

the

circ

um

fere

nce

of

NS

A s

oftb

alls

.

b

. F

ind

the

rati

o of

th

e vo

lum

e of

ML

B

base

ball

s to

th

e vo

lum

e of

NS

A

soft

ball

s. R

oun

d to

th

e n

eare

st t

enth

.

6

.75 i

n.

4

0 c

m

6

144 c

ub

ic f

eet

of

gra

in

1

.5:1

3

:4

2

7:6

4

045_

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GE

OC

RM

C12

_890

521.

indd

535/

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93:

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9P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 12

5

4

Gle

ncoe

Geo

met

ry

Enri

chm

ent

Do

ub

lin

g S

izes

Con

side

r w

hat

hap

pen

s to

su

rfac

e ar

ea w

hen

th

e si

des

of a

fig

ure

are

dou

bled

.

Th

e si

des

of t

he

larg

e cu

be a

re t

wic

e th

e si

ze o

f th

e si

des

of t

he

sm

all

cube

.

1. H

ow l

ong

are

the

edge

s of

th

e la

rge

cube

?

2. W

hat

is

the

surf

ace

area

of

the

smal

l cu

be?

3. W

hat

is

the

surf

ace

area

of

the

larg

e cu

be?

4. T

he

surf

ace

area

of

the

larg

e cu

be i

s h

ow m

any

tim

es g

reat

er

than

th

at o

f th

e sm

all

cube

?

Th

e ra

diu

s of

th

e la

rge

sph

ere

at t

he

righ

t is

tw

ice

the

radi

us

of

the

smal

l sp

her

e.

5. W

hat

is

the

surf

ace

area

of

the

smal

l sp

her

e?

6. W

hat

is

the

surf

ace

area

of

the

larg

e sp

her

e?

7. T

he

surf

ace

area

of

the

larg

e sp

her

e is

how

man

y ti

mes

gr

eate

r th

an t

he

surf

ace

area

of

the

smal

l sp

her

e?

8. I

t ap

pear

s th

at i

f th

e di

men

sion

s of

a s

olid

are

dou

bled

, th

e su

rfac

e ar

ea i

s m

ult

ipli

ed b

y .

Now

con

side

r h

ow d

oubl

ing

the

dim

ensi

ons

affe

cts

the

volu

me

of a

cu

be.

Th

e si

des

of t

he

larg

e cu

be a

re t

wic

e th

e si

ze o

f th

e si

des

of t

he

smal

l cu

be.

9. H

ow l

ong

are

the

edge

s of

th

e la

rge

cube

?

10. W

hat

is

the

volu

me

of t

he

smal

l cu

be?

11. W

hat

is

the

volu

me

of t

he

larg

e cu

be?

12. T

he

volu

me

of t

he

larg

e cu

be i

s h

ow m

any

tim

es g

reat

er t

han

th

at o

f th

e sm

all

cube

?

Th

e la

rge

sph

ere

at t

he

righ

t h

as t

wic

e th

e ra

diu

s of

th

e sm

all

sph

ere.

13. W

hat

is

the

volu

me

of t

he

smal

l sp

her

e?

14. W

hat

is

the

volu

me

of t

he

larg

e sp

her

e?

15. T

he

volu

me

of t

he

larg

e sp

her

e is

how

man

y ti

mes

gre

ater

th

an t

he

volu

me

of t

he

smal

l sp

her

e?

16. I

t ap

pear

s th

at i

f th

e di

men

sion

s of

a s

olid

are

dou

bled

, th

e vo

lum

e is

mu

ltip

lied

by

.

5 in

.

3 m

5 in

.

3 m

12-8

10 i

n.

150 i

n2

600 i

n2

4 t

imes

36

π m

2

144

π m

2

4 t

imes

4 10 i

n.

125 i

n3

1000 i

n3

8 t

imes

36

π m

3

288

π m

3

8 t

imes

8

045_

055_

GE

OC

RM

C12

_890

521.

indd

544/

10/0

89:

11:0

9P

M



Copyright

© G

lencoe/M

cG

raw

-Hill

, a d

ivis

ion o

f T

he M

cG

raw

-Hill

Com

panie

s,

Inc.

An

swer

s


PDF Pass

Chapter 12 Assessment Answer KeyQuiz 1 (Lessons 12-1 and 12-2) Quiz 3 (Lessons 12-4 and 12-5) Mid-Chapter TestPage 57 Page 58 Page 59

Quiz 4 (Lessons 12-7 and 12-8) Page 58

1.

2.

3.

4.

5.

116.1 in2

188.5 ft2

267.0 ft2

1152 ft3

2827.4 in3

1.

2.

3.

4.

5.

156 cm3

1005.3 cm3

261.8 cm3

201.1 m2

C

1.

2.

3.

4.

5.

� ⎯⎯ � WX and � ⎯ � VU , −−

ZX,

and � RSZ

yes

similar, 2:1

27:125

27:64

1.

2.

3.

4.

5.

C

G

D

F

D

6.

7.

8.

9.

10.

92.8 units2

2884.0 in2

150.8 in2

1.

2.

3.

4.

5.

8.6 m

120 units2

360 cm2

B

Quiz 2 (Lessons 12-3 and 12-4)

Page 57

A27-A36_GEOCRMC12_890521.indd 27A27-A36_GEOCRMC12_890521.indd 27 6/1/09 12:26:45 PM6/1/09 12:26:45 PM

Copyrig

ht ©

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cG

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-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

panie

s, In

c.


PDF Pass

Chapter 12 Assessment Answer KeyVocabulary Test Form 1 Page 60 Page 61 Page 62

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

slant height

similar solids

right cylinder

volume

axis

congruent solids

false; pyramid

true

true

false; oblique cone

the sum of the areas of the lateral

faces

a prism whose lateral edges are

also altitudes

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

C

F

B

G

D

H

A

F

B

H

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

G

D

H

D

F

B

G

B

J

B: 1020 ft2


Copyright

© G

lencoe/M

cG

raw

-Hill

, a d

ivis

ion o

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he M

cG

raw

-Hill

Com

panie

s,

Inc.

An

swer

s


Chapter 12 Assessment Answer KeyForm 2A Form 2B Page 63 Page 64 Page 65 Page 66

12.

13.

14.

15.

16.

17.

18.

19.

20.

H

C

J

B

G

B

H

C

G

B: 53 ft2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

D

H

D

G

C

F

C

F

B

G

C

12.

13.

14.

15.

16.

17.

18.

19.

20.

H

A

H

B

F

D

F

B

H

B: 391.6 ft2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

J

A

G

D

H

A

H

A

G

B

PDF 2ndA27-A36_GEOCRMC12_890521.indd 29A27-A36_GEOCRMC12_890521.indd 29 6/1/09 12:26:52 PM6/1/09 12:26:52 PM

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ht ©

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-Hill, a

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isio

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e M

cG

raw

-Hill C

om

panie

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c.


PDF Pass

Chapter 12 Assessment Answer KeyForm 2C Page 67 Page 68

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

2 ft

7690.6 ft3

467.7 ft3

103.2 ft3

1140.4 cm2

208.8 in3

great circle

neither

612.5 m2

113.1 in2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

�PQRS, �PQT,

�QTR, �RTS, �PTS

120 cm2

6 in2

524.1 yd2

6 in.

270 in2

270 + 150 √ � 3 in2

62.8 ft2

113.1 ft2


Copyright

© G

lencoe/M

cG

raw

-Hill

, a d

ivis

ion o

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he M

cG

raw

-Hill

Com

panie

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Inc.

An

swer

s


PDF Pass

Chapter 12 Assessment Answer KeyForm 2D Page 69 Page 70

12.

13.

14.

15.

16.

17.

18.

19.

576 in3

923.6 cm3

2057 in3

100.5 cm3

1592.8 in2

106.4 cm3

similar

1562.5 ft3

Lines in spherical

geometry are great

circles. Great circles

always intersect,

therefore they cannot be

parallel.

20.

B: 201.1 in2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

−−

GJ , −−

HJ , −

IJ , −−

GH , −−

HI , −−

GI

750 in2

258 ft2

845.2 m2

7 ft

540 ft2

931.0 ft2

204.2 cm2

282.7 cm2


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Chapter 12 Assessment Answer KeyForm 3 Page 71 Page 72

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

It is a rectangle.

128 + 4 √ �� 21

units2

35 + 7 √ �� 13 ft2

528 units2

17,837.8 ft2

32,435.2 ft2

468 in2

842.1 in2

233.8 in2

387.7 in2

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

184 cm3

8 m

No, it will overflow.

148.4 in3 > 96 in3

121.5 in3

3πr2

Yes; 33.5 cm3

< 37.7 cm3

similar

1: 3 √ � 4

700 ft2

No. Any three points

on a sphere determine

a triangle.

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Chapter 12 Assessment Answer KeyExtended-Response Test, Page 73

Scoring Rubric

Score General Description Specifi c Criteria

4 Superior

A correct solution that

is supported by well-developed,

accurate explanations

• Shows thorough understanding of the concepts of pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.

• Uses appropriate strategies to solve problems.

• Computations are correct.

• Written explanations are exemplary.

• Figures and drawings are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution, but may

contain minor fl aws in reasoning or

computation

• Shows an understanding of the concepts of pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.

• Uses appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are effective.

• Figures and drawings are mostly accurate and appropriate.

• Satisfi es all requirements of problems.

2 Nearly Satisfactory

A partially correct interpretation and/or

solution to the problem

• Shows an understanding of most of the concepts of

pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.

• May not use appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are satisfactory.

• Figures and drawings are mostly accurate.

• Satisfi es the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no supporting

evidence or explanation

• Final computation is correct.

• No written explanations or work shown to substantiate the

fi nal computation.

• Figures and drawings may be accurate but lack detail or

explanation.

• Satisfi es minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution indicating no

mathematical understanding of the

concept or task, or no solution is given

• Shows little or no understanding of most of the concepts of

pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.

• Does not use appropriate strategies to solve problems.

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Figures and drawings are inaccurate or inappropriate.

• Does not satisfy requirements of problems.

• No answer given.


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Chapter 12 Assessment Answer KeyExtended-Response Test, Page 73

Sample Answers

1. The lateral area is the area of the lateral faces. The surface area includes the area of the lateral faces plus the areas of the two bases.

2.

Oblique Right

3. Sample answer: Sam is painting the walls of a room. The room is 12 feet long, 10 feet wide, and 8 feet high. A gallon of paint covers 400 square feet and costs $16 per gallon. Find the cost of the paint needed to paint the room.

4. The first cylinder could have a radius of 3, a height of 4, a volume of 36π cubic units, and a surface area of 42π square units. The second cylinder could have a radius of 1, a height of 30, a volume of 30π cubic units, and a surface area of 62π square units.

5.

4 in.4 in.4 in.

12 in.

4 in.

4 in.

The volume of the pyramid is 16 • 12 − 3 or 64 cubic units and

the volume of the prism is 4 • 4 • 4 or 64 cubic units.

6. The volume of the cylinder is πr2 • 2r.

The volume of the hemisphere is 2πr3 −

3 .

The volume of the cone is πr2 • r − 3

Therefore, the total volume is

2πr3 + 2πr3

− 3 + πr3

− 3 or 3πr3 cubic units.

In addition to the scoring rubric found on page A33, the following sample answers may be used as guidance in evaluating open-ended assessment items.


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Chapter 12 Assessment Answer KeyStandardized Test Practice Page 74 Page 75

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

8. F G H J

A B C D

A B C D

9.

10.

11.

12. F G H J

A B C D

F G H J

A B C D

13.

14. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 4

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

5


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Chapter 12 Assessment Answer KeyStandardized Test Practice Page 76

15.

16.

17.

18.

19.

20.

21a.

b.

c.

d.

2.6 m

25

38.5 cm2

pentagonal

pyramid

3141.6 in2

2206.2 cm2

13,684.8 m2

207.3 m

3421.2 m2

10,263.6 m2


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Chapter 12 Resource Masters - Ms. Valeska...The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment