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ABC
DE F
GHI
JK L
edge
vertex
face
Copyright © by Holt, Rinehart and Winston. 141 Holt GeometryAll rights reserved.
Name Date Class
Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary.
Classifying Three-Dimensional FiguresClassify the figure. Name the vertices, edges, and bases.
What shape are the bases?
What shape are the faces?
Classify the figure.
Name the vertices.
Name the edges.
Name the bases.
Identifying a Three-Dimensional Figure From a NetDescribe the three-dimensional figure that can be made from the given net.
A. The net has two congruent faces that are .
The remaining faces are .
The net forms a .
B. The net has face that is a pentagon.
The remaining faces are .
The net forms a .
Describing Cross Sections of Three-Dimensional FiguresDescribe the cross section.
How many sides does the cross section
of this figure have?
What is the shape of the cross section of
the figure? .
Ready To Go On? Skills Intervention10-1 Solid Geometry10A
SECTION
Vocabulary
face edge vertex prism cylinder pyramid
cone cube net cross section
Find these vocabulary words in Lesson 10-2 and the Multilingual Glossary.
Drawing Orthographic Views of an ObjectDraw all six orthographic views of the given object. Assume there are no hidden cubes.
Orthographic drawings are drawings of the figure from six differentsides: top, bottom, front, back, left, and right. To draw orthographicviews, pretend you are looking at the figure from that view.
10AReady To Go On? Skills Intervention10-2 Representations of Three-Dimensional Figures
Copyright © by Holt, Rinehart and Winston. 142 Holt GeometryAll rights reserved.
Name Date Class
SECTION
Vocabulary
orthographic drawing isometric drawing perspective drawing
vanishing point horizon
Top
Bottom
Right
Left
Back
Front
Find this vocabulary word in Lesson 10-3 and the Multilingual Glossary.
Examining PolyhedronsFind the number of vertices, edges, and faces of a pentagonal pyramid. Use your results to verify Euler’s Formula.
Number of vertices: V �
Number of edges: E �
Number of faces: F �
Euler’s Formula states that V � E � F � 2.
Substitute the numbers into Euler’s Formula:
� � � 2
Simplify. � 2
Does your result verify Euler’s Formula?
Finding Distances and Midpoints in Three DimensionsFind the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary.
(5, 18, �12) and (�4, 12, �14)
distance:
d � ��� ( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ( z 2 � z 1 ) 2 Use the Distance Formula.
� ����
� �4 � � 2
� � � 18 � 2
� � �14 � � 2
Substitute known values.
� ���
� � 2
� � � 2
� � � 2
Simplify.
� ��
� � � ��
� Simplify.
midpoint:
M � x 1 � x 2 _______ 2 , y
1 � y 2 _______ 2 , z
1 � z 2 _______ 2 � Use the Midpoint Formula.
M � 5 � ________ 2 , � 12 _________ 2 , �12 � ____________ 2 � Substitute known values.
M � _____ 2 , _____ 2 , ______ 2 � � , , � Simplify.
10AReady To Go On? Skills Intervention10-3 Formulas in Three Dimensions
Name Date Class
Copyright © by Holt, Rinehart and Winston. 143 Holt GeometryAll rights reserved.
SECTION
Vocabulary
polyhedron
Copyright © by Holt, Rinehart and Winston. 144 Holt GeometryAll rights reserved.
When you move around, the distance you travel can be modeled on a three-dimensional coordinate plane.
A hiker stops for lunch at a picnic area 75 yards north and 30 yards east from her starting point. The elevation of the picnic area is 40 yards higher than the starting point. What is the distance from her starting point to the picnic area?
Understand the Problem
1. How can you use geometry to solve this problem?
2. What coordinate does each direction represent?
Make a Plan
3. What ordered triple represents the hiker’s starting point?
4. What ordered triple represents the picnic area?
5. State the Distance Formula.
Solve
6. Substitute the ordered triples into the Distance Formula and simplify.
d � ���
( � 0 ) 2 � ( � 0 ) 2 � (40 � ) 2
��
3 0 2 � 2 � 4 0 2 � ��
�
7. What is the distance from the hiker’s starting point to the picnic area?
Look Back
8. Find the length of the diagonal of a 30 yd by 75 yd by 40 yd rectangular prism.
9. Does this verify the answer you got in Exercise 6? Explain.
Ready To Go On? Problem Solving Intervention10-3 Formulas in Three Dimensions
Name Date Class
SECTION
10A
(30, 75, 40)
z
xy
Copyright © by Holt, Rinehart and Winston. 145 Holt GeometryAll rights reserved.
Name Date Class
10-1 Solid GeometryClassify each figure. Name the vertices, edges, and bases.
1. 2. 3.
AB
C
DE
F
L
M
H
I J
K
G
NM
L
Describe the three-dimensional figure that can be made from the given net.
4. 5. 6.
Describe each cross section.
7. 8. 9.
Ready To Go On? Quiz10ASECTION
Copyright © by Holt, Rinehart and Winston. 146 Holt GeometryAll rights reserved.All rights reserved.
10-2 Representations of Three-Dimensional FiguresUse the figure made of unit cubes for Exercises 10 and 11. Assume there are no hidden cubes.
10. Draw all six orthographic views. 11. Draw an isometric view.
12. Draw a cube in one-point 13. Draw a rectangle in two-point perspective. perspective.
10-3 Formulas in Three DimensionsFind the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula.
14. a square pyramid
15. a hexagonal prism
16. a triangular prism
17. A bird flies from its nest to a point that is 12 feet east, 9 feet south, and 8 feet higher in the tree than the nest. How far is the bird from the nest?
Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary.
18. (0, 0, 0) and (9, 30, 50) 19. (7, �5, 15) and (8, 3, �17) 20. (�1, 3, �1) and (�5, 7, 1)
10AReady To Go On? Quiz continued
Name Date Class
SECTION
Copyright © by Holt, Rinehart and Winston. 147 Holt Geometry
Ready To Go On? Enrichment10A
Name Date Class
SECTION
Three-Dimensional FiguresAnswer each question.
1. The cube at the right is built from 64 smaller cubes. The cube is spray painted on all sides.
How many of the cubes are painted on three faces?
How many cubes are painted on two faces?
How many cubes are painted on only one face?
2. A triangle has vertices J(4, 1, �3), K(�2, 6, �7), and L(10, �4, 1).
Classify the triangle by the length of its sides.
Find the perimeter of the triangle.
3. A polygon is called a Platonic Solid if all of its faces are regular polygons and if each vertex is the point of intersection of the same number of edges. The Platonic Solids are pictured at the right. Complete the table below. Verify that Euler’s Formula works for the Platonic Solids.
NameShape of
face
Number of edges at each vertex
Number of
Faces
Number of
Edges
Number of
Vertices
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
Does Euler’s Formula work for each Platonic Solid?
Tetrahedron
Icosahedron
Dodecahedron
Octahedron
Copyright © by Holt, Rinehart and Winston. 148 Holt Geometry
Ready To Go On? Skills Intervention10-4 Surface Area of Prisms and Cylinders
Find these vocabulary words in Lesson 10-4 and the Multilingual Glossary.
Finding Lateral Areas and Surface Areas of PrismsFind the lateral area and surface area of the rectangular prism. Round to the nearest tenth, if necessary.
lateral area:
L � Ph Use the formula for lateral area.
P � 2( ) � 2( ) � ft 2 P is the perimeter of the base.
L � ( )( ) Substitute known values.
L � ft 2 Simplify.
surface area:
S � L � 2B Use the formula for surface area.
B � ( )( ) � ft 2 B is the area of the base.
S � � 2( ) Substitute known values.
S � ft 2 Simplify.
Finding Lateral Areas and Surface Areas of Right CylindersFind the lateral area and surface area of the right cylinder. Give your answer in terms of �.
lateral area:
L � 2�r h Use the formula for lateral area.
L � 2�( )( ) Substitute known values.
L � mm2 Simplify.
surface area:
S � L � 2�r 2 Use the formula for surface area.
S � � 2�( )2 Substitute known values.
S � mm2 Simplify.
Name Date Class
SECTION
10B
Vocabulary
lateral face lateral edge right prism oblique prism
altitude surface area lateral surface axis of a cylinder
right cylinder oblique cylinder
4 ft 5 ft
6 ft
20 mm15 mm
Copyright © by Holt, Rinehart and Winston. 149 Holt GeometryAll rights reserved.
Find these vocabulary words in Lesson 10-5 and the Multilingual Glossary.
Finding Lateral Areas and Surface Areas of PyramidsFind the lateral and surface area of the square pyramid.
lateral area:
L � 1 __ 2 P� Use the formula for lateral area.
P � 4s � 32 in.2 P is the perimeter of the base.
L � 1 __ 2 ( )( ) Substitute known values.
L � in.2 Simplify.
surface area:
S � L � B
B � ( )2 � B is the area of the base of the pyramid.
S � � Substitute known values.
S � in. 2 Simplify.
Finding Lateral Areas and Surface Areas of Right ConesFind the lateral area and surface area of the right cone.
lateral area:
L � �r� Use the formula for lateral area of a cone.
L � �( )( ) Substitute known values.
L � � cm2 Simplify.
surface area:
S � L � B Use the formula for surface area of a cone.
B � �r 2 � �( )2 � 225� cm2 B is the area of the base of the cone.
S � � � � Substitue known values.
S � � cm2 Simplify.
10BReady To Go On? Skills Intervention10-5 Surface Area of Pyramids and Cones
Name Date Class
SECTION
Vocabulary
vertex of a pyramid regular pyramid slant height of a regular pyramid
altitude of a pyramid vertex of a cone axis of a cone right cone
oblique cone slant height of a right cone altitude of a cone
5 in.
8 in.
39 cm
15 cm
14 in.
8 in.6 in.
Copyright © by Holt, Rinehart and Winston. 150 Holt GeometryAll rights reserved.
Find this vocabulary word in Lesson 10-6 and the Multilingual Glossary.
Finding Volumes of PrismsFind the volume of the prism. Round to the nearest tenth, if necessary.
V � �wh Use the formula for volume of a right rectangular prism.
V � ( )( )( ) Substitute known values.
V � 280 ft3 Simplify.
Finding Volumes of CylindersFind the volume of the cylinder. Give your answer both in terms of � and rounded to the nearest tenth.
V � �r 2h Use the formula for volume of a cylinder.
V � �( )2( ) Substitute known values.
V � � cm3 Simplify. Leave your answer in terms of �.
V � cm3 Round your answer to the nearest tenth.
Exploring Effects of Changing DimensionsThe dimensions of a triangular prism are multiplied by 2. Describe the effect on the volume.
Volume using original dimensions:
V � Bh Use the formula for volume of a prism.
B � 1 __ 2 ( )( ) � in.3 B is the area of the base.
V � ( )( ) � 336 in.3 Substitute known values and simplify.
Volume after multiplying each dimension by 2:
V � Bh Use the formula for volume of a prism.
B � 1 __ 2 ( )( ) � in.3 B is the area of the base.
V � ( )( ) � 2688 in.3 Substitute known values and simplify.
When the dimensions of a triangular prism are multiplied by 2, the volume
is multiplied by or .
10BReady To Go On? Skills Intervention10-6 Volume of Prisms and Cylinders
Name Date Class
SECTION
Vocabulary
volume
12 cm
8 cm
7 ft10 ft
4 ft
12 in.28 in.
16 in.
Copyright © by Holt, Rinehart and Winston. 151 Holt Geometry
If you know the volume and the density of a three-dimensional object, you can find its weight.
A stone fireplace measures 6 m by 4 m by 11 cm. Find the volume of the fireplace. If the density of stone is 2515 kilograms per cubic meter, what is the mass of the fireplace in kilograms?
Understand the Problem
1. What two measurements must you calculate?
2. How many meters is 11 centimeters?
3. What conversion factor will you use to calculate the mass of the fireplace?
Make a Plan
4. What formula will you use to find the volume of the fireplace?
5. After you find the volume, how will you find the mass?
Solve
6. Calculate the volume of the fireplace.
7. Multiply your answer from Exercise 6 by the conversion factor from Exercise 3.
8. What is the mass of the fireplace in kilograms?
Look Back
9. Use this formula to check your solution: density � mass ______ volume
Substitute the values you found for the mass and volume.
density � mass ______ volume � __________ 2.64 �
10. Does your answer check?
10BReady To Go On? Problem Solving Intervention10-6 Volume of Prisms and Cylinders
Name Date Class
SECTION
Copyright © by Holt, Rinehart and Winston. 152 Holt Geometry
Finding Volumes of PyramidsFind the volume of the pyramid.
V � 1 __ 3 Bh Use the formula for volume of a pyramid.
B � ( )2 � 49 yd2 Find B, the area of the base of the pyramid.
V � 1 __ 3 ( )( ) Substitute known values.
V � yd3 Simplify.
Finding Volumes of ConesFind the volume of the cone. Give your answer both in terms of � and rounded to the nearest tenth.
V � 1 __ 3 Bh Use the formula for volume of a cone.
V � 1 __ 3 �r 2h B is the area of a circle.
V � 1 __ 3 �( )2( ) Substitute known values.
V � � � ____ 3 � � � in.3 Simplify. Leave your answer in terms of �.
V � in.3 Round your answer to the nearest tenth.
Finding Volumes of Composite Three-Dimensional FiguresFind the volume of the composite figure. Round to the nearest tenth.
Find the volume of the cube.
V � s3 Use the formula for volume of a cube.
V � ( )3 Substitute and simplify.
V � cm3
Find the volume of the square pyramid.
V � 1 __ 3 Bh Use the formula for volume of a pyramid.
V � 1 __ 3 ( )2( ) Substitute and simplify.
V � _____ 3 cm3
Subtract the volume of the pyramid from the volume of the cube to find the volume of the composite figure.
V composite � � _____ 3 � _____ 3 � cm3
Ready To Go On? Skills Intervention10-7 Volume of Pyramids and Cones
Name Date Class
SECTION
10B
5 cm 5 cm
5 cm
11 in.
6 in.
9 yd
7 yd7 yd
Copyright © by Holt, Rinehart and Winston. 153 Holt GeometryAll rights reserved.
Name Date Class
Find these vocabulary words in Lesson 10-8 and the Multilingual Glossary.
Finding Volumes of SpheresFind each measurement. Give your answer in terms of �.
A. the volume of the sphere
V � 4 __ 3 �r 3 Use the formula for volume of a sphere.
V � 4 __ 3 �( )3 Substitute 15 for r.
V � � m3 Simplify.
B. the volume of the hemisphere
r � 1 __ 2 d � 1 __ 2 ( ) � Find the radius of the hemisphere.
V � 2 __ 3 �r 3 Use the formula for volume of a hemisphere.
V � 2 __ 3 �( )3 Substitute known values.
V � � yd3 Simplify.
Finding Surface Areas of SpheresFind the surface area of the sphere. Give your answer in terms of �.
S � 4�r 2 Use the formula for surface area of a sphere.
S � 4�( )2 Substitute known values.
S � � in.2 Simplify.
Exploring Effects of Changing Dimensions
The radius of the sphere is multiplied by 1 __ 2 . Describe the effect on the volume.
original dimensions: radius multiplied by 1 __ 2 :
V � 4 __ 3 � r 3 V � 4 __ 3 � r 3
V � 4 __ 3 �( ) 3 � ________ 3 � f t 3 V � 4 __ 3 �( ) 3 � ______ 3 � f t 3
If the radius is multiplied by 1 __ 2 , the volume is multiplied by � � 3
or 1 ____
.
Ready To Go On? Skills Intervention10-8 Spheres10B
SECTION
Vocabulary
sphere center of a sphere radius of a sphere
hemisphere great circle
10 ft
24 yd
15 m
8 in.
Copyright © by Holt, Rinehart and Winston. 154 Holt GeometryAll rights reserved.
Changing the dimensions of a three-dimensional object affects the volume of the object.
A scale model of the solar system has a sphere with radius 3.4 cm to represent Mars and a sphere with radius 6.4 cm to represent Earth. About how many times as great is the volume of Earth as the volume of Mars?
Understand the Problem
1. What does “scale model” mean?
2. What measurements must you find?
Make a Plan
3. What formula will you use to calculate the volumes of the spheres?
4. After you find the volumes of the spheres, what will you do next?
Solve
5. Find the volume of the sphere representing Earth.
6. Find the volume of the sphere representing Mars.
7. Divide your answer from Exercise 5 by your answer from Exercise 6.
8. About how many times greater is the volume of the Earth?
Look Back
9. How many times greater is the radius of the Earth than the radius of Mars?
10. Cube your answer to Exercise 9.
11. Does your answer to Exercise 10 match your answer to Exercise 8?
10BReady To Go On? Problem Solving Intervention10-8 Spheres
Name Date Class
SECTION
3.4 cm
6.4 cm
Copyright © by Holt, Rinehart and Winston. 155 Holt GeometryAll rights reserved.
Ready To Go On? Quiz
10-4 Surface Area of Prisms and CylindersFind the surface area of each figure. Round to the nearest tenth, if necessary.
1. 2. 3.
4 in.
5 in.7 in.
5 m
9 m
3 cm3 cm
6 cm
3 cm
4. The dimensions of an 8 yd by 14 yd by 10 yd right rectangular prism are
multiplied by 3 __ 2 . Describe the effect on the surface area.
10-5 Surface Area of Pyramids and ConesFind the surface area of each figure. Round to the nearest tenth, if necessary.
5. a regular hexagonal pyramid with base edge length 9 ft and slant height 14.3 ft
6. a right cone with diameter 22 mm and height 13 mm
7. the composite figure formed by two pyramids shown at right
10-6 Volume of Prisms and CylindersFind the volume of each figure. Round to the nearest tenth, if necessary.
8. a regular pentagonal prism with base area 28 in.2 and height 7 in.
9. a cylinder with radius 12 yd and height 15 yd
10B
Name Date Class
SECTION
8 ft
8 ft8 ft
10. A brick patio measures 18 ft by 16 ft by 6 in. Find the volume of the bricks. If the density of the bricks is 130 pounds per cubic foot, what is the weight of the patio in pounds?
11. The dimensions of a hexagonal prism with base area 374 cm 2 and height 18 cm are tripled. Describe the effect on the volume.
10-7 Volume of Pyramids and ConesFind the volume of each figure. Round to the nearest tenth, if necessary.
12. 13. 14.
12 cm
17 cm
12 m
27 m14 m
22 yd
11 yd
10-8 SpheresFind the surface area and volume of each figure. Give your answers in terms of �.
15. a sphere with diameter 26 in.
16. a hemisphere with radius 7 m
17. A junior basketball has a diameter of approximately 7 in., and a regulation basketball has a diameter of approximately 9.5 in. About how many times as great is the volume of the regulation basketball as the volume of the junior basketball?
Copyright © by Holt, Rinehart and Winston. 156 Holt GeometryAll rights reserved.
Name Date Class
Ready To Go On? Quiz continued
10BSECTION
10BReady To Go On? EnrichmentSurface Area and Volume
Answer each question.
1. The volume of a sphere has the same measure in cubic units as does its surface area in square units. What is the radius of the sphere?
2. A backyard swimming pool measures 10 ft by 14 ft and is 5 ft deep. The pool has a leak and is losing water at the rate of 4 in. per day. If the owner does not refill the water, in how many days will the pool be empty?
3. In the figure at the right, if you rotate the rectangle around the dashed line, the resulting figure will be a cylinder with radius 3 cm and height 7 cm.
Find the surface area of the cylinder. Leave your answer in terms
of �.
Find the volume of the cylinder.
4. What figure results from rotating the figure at the right around
the dashed line?
Find the surface area of the figure.
Find the volume of the figure.
5. The figure at the right is an octahedron. Each face is an equilateral triangle. The length of each edge is 18 in.
Find the surface area of the octahedron.
Find the volume of the octahedron.
6. Daniel is repainting the walls in his room. His room is 15 ft wide, 12 ft long and 8 ft high. He plans to put two coats of paints on each wall. One gallon of paint covers approximately 400 ft 2. How many gallons of paint must Daniel buy? Explain your answer.
5 m13 m
7 cm
3 cm
18 in.
Name Date Class
Copyright © by Holt, Rinehart and Winston. 157 Holt GeometryAll rights reserved.
SECTION