Introduction to the Third Dimensionmrpack.weebly.com/uploads/8/5/0/5/8505687/course_1_chapter_14.… · Sketch.various.views.of.a.solid.figure.to.provide.a. two-dimensional.representation.of.a.three-dimensional.figure

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Introduction to the Third Dimension

14.1 Cut, Fold, and Voila!Nets........................................................................... 905

14.2 More Cans in a CubeThe.Cube......................................................................921

14.3 Prisms Can Improve Your Vision!Prisms........................................................................ 941

14.4 Outside and Inside a PrismSurface.Area.and.Volume.of.a.Prism............................953

14.5 The Egyptians Were on to Something—or Was It the Mayans?Pyramids.....................................................................963

14.6 And The Winning Prototype Is . . . ?Identifying.Geometric.Solids..

in.Everyday.Occurrences..............................................979

Rice is central to

the daily diet of billions of people around the world. Ornamental

containers, such as the ceramic canister shown, are common in kitchens where rice

is cooked.

904. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension

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14.1. . . Nets. . . •. . . 905

Have.you.ever.heard.of.the.term.rebranding?.Generally,.this.term.means.to.

give.a.product.or.an.item.a.new.look..Rebranding.isn’t.a.decision.that.businesses.

take.lightly..Many.times,.marketing.research.is.performed.on.a.product’s.current.

look.and.possible.new.looks..There.is.also.the.risk.that.people.will.not.recognize.

the.product,.perhaps.leading.to.fewer.sales..What.items.or.products.have.you.

seen.that.have.gone.through.rebranding?.Do.you.think.rebranding.only.deals.with.

products.or.items?

Cut, Fold, and Voila!Nets

Key Terms. geometric.solids

. prototype

. edge

. face

. vertex

. net

Learning GoalsIn this lesson, you will:

. Sketch.various.views.of.a.solid.figure.to.provide.a.

two-dimensional.representation.of.a.three-dimensional.figure.

. Construct.a.net.from.a.model.of.a.geometric.solid.

. Construct.a.model.of.a.geometric.solid.from.a.net.

. Use.nets.to.provide.two-dimensional.representations.of.a.

geometric.solid.


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Problem 1 Prototype #1

Geometric solids are all bounded three-dimensional geometric figures. The three

dimensions are length, width, and height.

The new marketing director of a rice distribution company, Rice Is Nice, has decided

to change the way its product is packaged. The marketing director hopes to get more

people to notice and talk about the product. She assigned her product development team

to create prototypes. A prototype is a working model of a possible new product. Each

prototype needs to be a different-shaped container to package the product.

Rice is Nice is considering changing the dimensions of its current packaging. The box

shown is one prototype.

Prototype#1

The height of the box is 5.7 centimeters, the width or depth of the box is 2.9 centimeters,

and the length of the box is 4.3 centimeters.

1. Use the figure shown to answer each question.

a. How many sides of the box can you see?

b. Describe the location of the sides you can see.

c. How many sides can you not see?

d. What sides can you not see?

e. What is the shape of each side?

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14.1. . . Nets. . . •. . . 907

2. Sketch each side of the box, label the location of the side, and include the measurements.

Imagine cutting out each side you sketched and taping

the corresponding edges together to construct the box of

Prototype #1. An edge is the intersection of two faces of

a three-dimensional figure. A face is one of the polygons

that makes up a polyhedron. The point where edges meet is

known as a vertex of a three-dimensional figure.

A net is a two-dimensional representation of a three-dimensional

geometric figure. A net is cut out, folded, and taped to create a model of

a geometric solid.

A vertex of a solid is similar to the vertex of

an angle.

A net has all these properties:

● The net is cut out as a single piece.

● All of the sides of the geometric solid are

represented in the net.

● The sides of the geometric solid are drawn

such that they share common edges.

● The common edges are labeled

as fold lines.

● Tabs are drawn on the edges to

be taped.


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14.1. . . Nets. . . •. . . 909

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3. Cut, fold, and tape this net to create Prototype #1.

Prototype #1


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14.1. . . Nets. . . •. . . 911


Sandy created this net to model her prototype for rice packaging. Cut out, fold, and tape

this net to create a prototype.

Prototype #2


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14.1. . . Nets. . . •. . . 913


Emilia created this net to model her prototype for rice packaging. Cut out, fold, and tape


Prototype #3


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14.1. . . Nets. . . •. . . 915


Trang created this net to model his prototype for rice packaging. Cut out, fold, and tape


Pro

toty

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#4


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Hang on to your prototype

models, you will use these again in other lessons in

this chapter.

14.1. . . Nets. . . •. . . 917

Problem 5 Using Your Sorting Hat!

The marketing director requires the team members to present their prototypes at the next

Rice Is Nice stockholders’ meeting. She told the team that 4 prototypes are too many.

The team members could not decide which of the prototypes to exclude, so they intend to

group the 4 prototypes into 2 categories and highlight each category.

1. Using all 4 solids, sort them into 2 groups, and explain your reasoning.

2. Compare your method of grouping with your classmates. Did everyone use the same

groupings? Explain your reasoning.


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The net for each prototype you created is shown. You will use these representations again

in this chapter.

Pro

toty

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#1

5.7 cm

2.9 cm

4.3

cm

Prototype #2

4.3 cm

4.3 cm

5.7

cm

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14.1. . . Nets. . . •. . . 919

Prototype #3

3.8 cm

5.3 cm

Prototype #4

3.3

cm

5.1 cm

3.7 cm

Be prepared to share your solutions and methods.

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14.2. . . The.Cube. . . •. . . 921

Only.90.years.ago,.the.standard.beverage.size.bottle.was.6.5.ounces..Now,.

some.convenient.stores.tout.128-ounce.drinks!.But.the.amount.in.one.serving.

isn’t.the.only.thing.that.has.gotten.bigger..Why.do.you.think.drink.portions.have.

become.larger.over.the.years?.Do.you.think.the.common.practice.of.restaurants.

refilling.drinks.is.a.contributing.factor?

More Cans in a CubeThe Cube


. Create.a.model.of.a.cube.from.a.net.

. Construct.a.model.of.a.geometric.solid.from.a.net.

. Use.nets.to.provide.two-dimensional.representations.of.

a.cube.

. Estimate.the.volume.and.surface.area.of.a.cube.

. Use.nets.to.compute.the.volume.and.surface.area.of.a.

cube.

. Use.a.formula.to.determine.the.volume.of.a.cube.

. Use.unit.cubes.to.estimate.the.surface.area.and.volume.

of.larger.cubes.

. Use.appropriate.units.of.measure.when.computing.the.

surface.area.and.volume.of.a.cube.

. Explore.how.doubling.the.dimensions.of.a.cube.affects.

the.volume.of.the.cube.

Key Terms. point

. line.segment

. polygon

. polyhedron

. regular.polyhedron

. congruent

. cube

. unit.cube

. diameter

. surface.area

. volume


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Problem 1 Speaking a Common Language

Before beginning the lesson, everyone must speak a common language. It is important

to use the same words when studying mathematics and describing geometric terms.

For example, a word you may have used in the past may actually have a more precise

definition when dealing with mathematics. For example, the word point has many

meanings outside of math. However, the mathematical definition of point is a location

in space. A point has no size or shape, but it is often represented by using a dot and is

named by a capital letter. A line segment is a portion of a line that includes two points

and all the points between those two points.

Recall, a polygon is a closed figure formed by three or more line segments. Knowing

these definitions will help you learn the meanings of other geometric words.

1. What do you think is the meaning of a closed figure?

2. Sketch what you think is an example of a polygon.

3. Is your sketch a closed figure? Are all of the sides in your sketch formed by line segments?

4. Compare your sketch with your classmates’ sketches. Did everyone sketch the same

polygon? Explain how your classmates’ and your sketches are the same or different.

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14.2. . . The.Cube. . . •. . . 923

A polyhedron is a three-dimensional figure that has polygons

as faces.

5. Sketch what you think is an example of a polyhedron.

6. Does your sketch look like a three-dimensional figure?

Does your sketch show polygons for every face?


polyhedron? Explain how the sketches are the same or different.

Would any of the prototypes

you created in the last lesson be

polyhedrons?

924 • Chapter 14 Introduction to the Third Dimension

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A regular polyhedron is a three-dimensional solid that has congruent regular polygons

as faces and has congruent angles between all faces. Congruent means having the same

size, shape, and measure.

8. Sketch what you think is an example of a regular polyhedron.

9. Does your sketch look like a three-dimensional solid that has congruent regular

polygons as faces and congruent angles between all the faces?


regular polyhedron? Explain how the sketches are the same or different.

A cube is a regular polyhedron whose six faces are congruent squares.

A unit cube is a cube that is one unit in length, one unit in width, and one unit in height.

In this chapter, unit cubes are used as manipulatives to explore characteristics of

geometric solids. The unit cubes are typically 1 centimeter in length, 1 centimeter in width,

and 1 centimeter in height. For this reason, use a centimeter ruler to measure lengths in

this chapter.

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14.2. . . The.Cube. . . •. . . 925

Problem 2 Is It Really a Cube?

In 1993, beverage manufacturers decided to repackage their product to boost sales.

Research indicated that consumers would rather buy more cans of their favorite

beverages at one time than make several trips to the store. The marketing team came up

with the idea of packaging several cans of their beverage together in a way that was easy

to carry. This packaging is called the “cube.”

A cube contains 24 cans.

1. Sketch some of the possible rectangular arrangements of 24 cans. Your arrangements

may have more than one layer.

The diameter of each can is 2 inches. The diameter is the distance across a circle through

its center. The height of each can is 6 inches.

2. What are the approximate dimensions of rectangular boxes needed to contain each

arrangement of cans you sketched in Question 1?


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3. The manufacturer decided to go with a two layer arrangement and called it a cube.

a. What are the dimensions of this arrangement?

b. Why do you think they made the decision to call this a cube?

c. Explain why calling the package a cube can be confusing.

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14.2. . . The.Cube. . . •. . . 927

Problem 3 Characteristics of a Cube

The cube is a basic geometric solid.

1. Sketch a cube.

2. How many faces of the cube can you see?

3. Describe the location of the faces you can see.

4. How many faces can you not see?

5. Describe the location of the faces that you cannot see.

6. What is known about the length, height, and width of the cube?


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7. Would you measure the length, width, and height of a cube using linear units such as

inches, centimeters, and feet? Or, would you use square units such as square inches,

square centimeters, and square feet? Or, would you use cubic units such as cubic

inches, cubic centimeters, and cubic feet?

8. Sketch and describe the shape of each face of a cube.

9. Is a cube a polygon? Explain your reasoning.

10. Is a cube a polyhedron? Explain your reasoning.

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14.2. . . The.Cube. . . •. . . 929

Problem 4 Cube Net

There are 11 different nets that can be created to model a cube.

1. Here is one example of a net of a cube.

Cut it out, fold it, and tape it together to create a geometric model of a cube.


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14.2 The Cube • 931

2. Describe the number of faces, vertices, and edges of the cube.

3. Not all of these nets create a cube. Circle each figure that is a net of a cube.

4. How did you determine which nets were cubes in Question 3?

Remember, a cube has six faces_and only

six faces!


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Problem 5 Surface Area of a Cube

Surface area is the total area of the two-dimensional surfaces (faces and bases) that

make up a three-dimensional object.

Consider the model of the cube you created in Problem 4, Question 1, to answer

each question.

1. What is true about the area of each of the 6 faces of a cube?

2. Is the area of a face of a cube measured using linear units such as inches,

centimeters, and feet? Or, using square units such as square inches, square

centimeters, and square feet? Or, using cubic units such as cubic inches, cubic

centimeters, and cubic feet?

3. Describe a strategy that you can use to determine the total surface area of a cube?

4. Use a centimeter ruler to calculate the total surface area of your cube.

5. How is the net of a cube helpful when determining the surface area of a cube?

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14.2. . . The.Cube. . . •. . . 933

6. What is the surface area of a unit cube?

7. How are unit cubes helpful when determining the

surface area of a larger cube?

Problem 6 Volume of a Cube

Volume is the amount of space occupied by an object.

The volume of a cube is calculated by multiplying the length times the width times the

height of the cube.

Use the model of the cube you created in Problem 4, Question 1, to answer each question.

1. Would you measure the volume of a cube using linear units such as inches,

centimeters, and feet? Or, would you use using square units such as square inches,

square centimeters, and square feet? Or, would you use using cubic units such as

cubic inches, cubic centimeters, and cubic feet?

2. How is estimating the volume of a cube different from calculating the volume of a cube?

Do you have unit cubes

ready?


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3. Estimate the volume of your cube by stacking unit

cubes next to the model of the cube.

4. Measure the length, width, and height of the cube using a

centimeter ruler. Then, multiply the length, width, and height to

calculate the volume.

5. What is the difference between the estimation of the volume and

the calculation of the volume?

6. What is the ratio of the difference between the estimation and the

calculation of the volume to the calculation of the volume?

7. Write the ratio from Question 6 as a percent. This is the percent of increase or

decrease in volume resulting from estimation.

8. How is the net of a cube helpful when determining the volume of a cube?

9. What is the volume of a unit cube?

10. How could unit cubes be helpful when you are determining the volume of a larger cube?

“Grab a handful of unit cubes!

What percent difference is considered a good

estimate?

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14.2. . . The.Cube. . . •. . . 935

Problem 7 Volume Formula

Volume can be determined by using the formula V 5 3 w 3 h, where V is the volume

of the cube, is the length of the cube, w is the width of the cube, and h is the height of

the cube.

The base of a cube is a square. Recall that the area of a square is calculated by

multiplying the length of the square by the width of the square. Written as a formula, the

area of the base of a cube is Area of the Base 5 3 w.

Consider the two formulas:

V 5 3 w 3 h

Area of the Base 5 3 w

If B is used to represent the area of the base of a cube, then

you can rewrite the second formula as: B 5 3 w.

Now consider the two formulas:

V 5 3 w 3 h

B 5 3 w

Using both of these formulas, you can rewrite the formula for

the volume of a cube as V 5 B 3 h, where V represents the

volume of the cube, B represents the area of the base of the

cube, and h represents the height of the cube. You can use this

formula to calculate the volume of many different geometric

solids. However, the formula for calculating the value of B will

change depending on the base shape of the polyhedron.


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Use the formula V 5 B 3 h to answer each question.

1. The length, width, and height of the cube are each equal to 2 centimeters.

2 cm

2 cm

2 cm

a. Calculate the area of the base of the cube.

b. What is the height of the cube?

c. Calculate the volume of the cube.

2. The volume of this cube is 27 cubic centimeters.

27 cm3

a. What is the area of the base of the cube?

b. What is the height of the cube?

“Keep in mind that a number doesn't say much without

a label.

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14.2. . . The.Cube. . . •. . . 937

Problem 8 Jerome and Roberta Need Your Help!

Jerome began stacking unit cubes to make a larger cube but was interrupted before he

could finish. The figure shown displays how much progress Jerome made in making the

larger cube.

1. You can see how long, wide, and tall Jerome wanted the cube. Calculate the volume

and surface area of Jerome’s cube if he had completed it.

Roberta is using unit cubes to determine the surface area and volume of larger cubes. She

wants to build 6 different size cubes to compare the surface area and volume, but she

realized that she would not have enough unit cubes to complete the models. She decides

to just build the length, width, and height of the first four cubes and to look for a pattern.

The figures Roberta built are shown.


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2. Roberta thinks she sees a pattern, but she needs to sketch the fifth and sixth cube to be

sure. Help Roberta by sketching the fifth and six figures based on the pattern you see.

3. Roberta is organizing the data in a table. Help Roberta complete the table.

Dimensions of the Cube

Area of One Side of the Cube

(in square units)

Surface Area of the Cube

(in square units)

Volume of the Cube

(in cubic units)

1.3.1.3.1

2.3.2.3.2

3.3.3.3.3

4.3.4.3.4

5.3.5.3.5

6.3.6.3.6

4. Describe how Roberta can use the dimension (length or width) of a cube to determine

the area of one side of the cube.

5. Describe how Roberta can use the area of one side of the cube to determine the

surface area of the cube.

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14.2. . . The.Cube. . . •. . . 939

6. Describe how Roberta can use the dimensions (length, width, and height) of the cube

to determine the volume of the cube.

7. Roberta is looking at the completed table and notices that when the dimensions of a

cube are doubled, the volume of the larger cube is predictable. She saw the pattern!

Describe the pattern Roberta sees in the completed table.

8. Use Roberta’s pattern and the volume of a 5 3 5 3 5-unit cube to predict the volume

of a 10 3 10 3 10-unit cube.

9. Use Roberta’s pattern and the volume of a 10 3 10 3 10-unit cube to predict the

volume of a 20 3 20 3 20-unit cube.

10. Roberta’s lab partner, Derrick, looked at her completed table in Question 3 and found it

interesting that the surface area and the volume of a 6 3 6 3 6-unit cube is 216. Roberta

helped Derrick understand that they were not equal. What did Roberta tell Derrick?

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Talk the Talk

Each numerical answer describes the volume or the surface area of a cube. Which is it?

How do you know?

1. 125

5

2. 24

2

3. 13.5 m2

4. 3.375 m3


“Labels really do matter.

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14.3. . . Prisms. . . •. . . 941

Key Termsprism

bases.of.a.prism

lateral.faces

height.of.a.prism

rectangular.prism

right.prism


. Sketch.a.model.of.a.right.rectangular.prism.

. Create.models.of.various.prisms.

. Determine.the.characteristics.of.various.prisms.

Do.you.know.that.binoculars.and.prisms.are.close.friends?.In.1854,.Ignazio.

Porro.realized.this.and.patented.the.“Porro.Prism.”.Using.right.triangular.prisms.

he.was.able.to.turn.an.image.right-side.up..Basically,.when.an.image.is.gathered.

by.the.lens,.the.image.is.upside.down..Thus,.inside.each.eyepiece.are.two.prisms,.

which.turn.the.image.right-side.up.and.invert.the.image.from.left.to.right..What.

do.you.think.would.happen.if.only.one.prism.was.used.in.binoculars?.What.do.you.

think.would.happen.if.4.prisms.were.used.in.binoculars?

Prisms Can Improve Your Vision!Prisms

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Problem 1 Getting to Know Prisms!

A prism is a polyhedron with two parallel and congruent faces and all the other faces

are parallelograms. These two parallel and congruent faces are known as the bases of a

prism. The remaining parallelogram-shaped faces are known as lateral faces.

1. What do you think “two parallel and congruent faces” means?

2. What is a parallelogram?

3. Sketch what you think is an example of a prism.

4. Does your sketch show two bases that are the same polygon and

are they drawn parallel to each other? Are all of the faces of your

polyhedron parallelograms except for the bases?

5. Identify the bases in your sketch.

Look at the prototype models

you created earlier for ideas.

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14.3. . . Prisms. . . •. . . 943


prism? Explain how the sketches are the same or different.

A height of a prism is the length of a line segment that is drawn from one base to the

other base. This line segment must be perpendicular to the other base.

7. Use your sketch to explain what is meant by “height of a prism”?

A rectangular prism is a prism that has a rectangle as its base.

8. Sketch what you think is an example of a rectangular prism.

9. Are all of the faces of your sketch rectangles? Does your sketch have two bases that

are parallel and congruent?


rectangular prism? Explain how the sketches are the same or different.

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A right prism is a prism that has bases aligned one directly above the other and has

lateral faces that are rectangles. All prisms associated with this chapter are right prisms.

11. Which faces of a prism are considered “lateral faces”?

12. Sketch what you think is an example of a right prism.

13. Are the bases of your prism aligned one directly above the other? Are all of the lateral

faces in your sketch rectangles?


right prism? Explain how the sketches are the same or different.

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14.3. . . Prisms. . . •. . . 945

Problem 2 Right Rectangular Prism

1. Use the model of the cube you created in Lesson 14.2 to answer each question.

a. Is a cube a prism? Explain your reasoning.

b. Is a cube a rectangular prism? Explain

your reasoning.

c. Is a cube a right prism? Explain your reasoning.

2. In your own words, describe what makes up a right rectangular

prism.

Can a square be a rectangle? Can a rectangle be a

square? Ah! My head is spinning!

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3. Sketch a right rectangular prism that is not a cube.

a. How many faces can you see?

b. Describe the location of the faces you can see.

c. How many faces can you not see?

d. Describe the location of the faces you cannot see.

e. What is known about the length, width, and height of the right rectangular prism?

f. Describe the shape of each face of the right rectangular prism.

“Have you created any other

model that is a right rectangular prism?

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14.3 Prisms • 947

Problem 3 Characteristics of a Prism

Use pasta and miniature marshmallows to construct a

model of each prism shown. Use your model to answer

questions about the prisms.

1. Construct and analyze this prism.

a. Name the polygon that is the base of this prism.

b. How many faces of the prism are lateral faces?

c. Identify the number of vertices, edges, and faces.

d. How is a height of this prism determined?

You can break the pasta

to be any length you want.


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3. Construct and analyze this prism





14.3 Prisms • 949

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14.3. . . Prisms. . . •. . . 951

5. Complete the table shown with the data from Questions 1, 2, 3, and 4.

Shape of the Base of Prism

(Regular Polygon)

Number of Sides

of the Base

Number of Vertices

Number of Edges

Number of Faces

6. Use the data from the table in Question 5 and any patterns you notice to answer

each question.

a. What is the relationship between the number of sides of the base and the number

of vertices of each prism?

b. What is the relationship between the number of sides of the base and the number

of edges of each prism?

c. What is the relationship between the number of sides of the base and the number

of faces of each prism?

7. Without making a model or drawing a sketch, predict the number of vertices, edges,

and faces for an octagonal prism. Describe your reasoning for making the prediction.

8. To verify your prediction, make a model of an octagonal prism to check your answers.


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14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 953

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Many.zoos.and.aquariums.that.hold.large.marine.animals.must.order.or.build.

custom-made.tanks.for.their.creatures..These.tanks.come.in.many.different.

shapes.and.sizes..A.large.fish.tank.that.might.hold.jellyfish.or.sea.horses.at.the.

zoo.might.have.dimensions.of.48"..24"..25".and.could.hold.around.115.gallons.

of.water..Tanks.for.large.animals.such.as.dolphins.and.whales.could.have.

dimensions.of.46'..23'..30'..That.means.these.tanks.hold.around.238,000.

gallons.of.water!.How.do.you.think.the.zoo.keepers.determine.what.size.the.tank.

should.be.and.how.much.water.it.should.hold?

Outside and Inside a PrismSurface Area and Volume of a Prism


. Use.unit.cubes.to.estimate.the.volume.and.surface.area.of.a.right.rectangular.prism.

. Use.nets.to.compute.the.volume.and.surface.area.of.a.right.rectangular.prism.

. Use.a.formula.to.determine.the.volume.of.a.right.rectangular.prism.

. Use.appropriate.units.of.measure.when.computing.the.surface.area.and.volume.of.a.

right.rectangular.prism.

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Problem 1 Right Rectangular Prism Net

1. The net shown is Prototype 1 from the first lesson.

5.7 cm

2.9 cm4.

3 cm Prototype

#1

a. Write the name of each side on each face: front, back, top,

bottom, left side, and right side.

b. Use the net to estimate the surface area of the right rectangular

prism. Recall that the unit of measurement when calculating the

surface area is square units.

c. Calculate the surface area of the right rectangular prism.

Explain your calculation.

“Get out your prototype 1 model to help answer

questions in this lesson.

Pairs of faces in the prism

are congruent. This can help me

estimate.

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d. How does the estimation of the surface area compare to the calculation of the

surface area?

e. Use your model of a right rectangular prism to determine the number of faces,

vertices, and edges.

f. Estimate the maximum number of unit cubes that would fit inside your model of a

right rectangular prism.

Calculating the actual volume of a right rectangular prism is similar to calculating the

actual volume of a cube. Multiply the length of the rectangular prism times the width of

the rectangular prism times the height of the rectangular prism, or calculate the area of the

base and multiply the product by the height.

g. Calculate the volume of the right rectangular prism. Recall that the unit of

measurement when calculating the volume is cubic units.

h. How does the estimation of the volume compare to the calculation of the volume?

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2. Place your model of the right rectangular prism you created on your desk such that it

rests on one of the largest sides.

a. Lightly sketch a letter X on the two bases of the prism.

b. Now, turn your model of the right rectangular prism such that it rests on one of the

smallest sides and lightly sketch a letter X on the two bases of the prism.

c. Are there two Xs on any face of the prism?

d. Think of the front, back, left, right, bottom, and top faces as locations on the prism.

Is the location of the bases in part (a) the same location as the bases in part (b)?

e. In your own words, describe how you can determine which sides of a right

rectangular prism are bases.

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Problem 2 Surface Area of a Prism

The surface area of the cube was calculated by adding all of the areas of all of the faces of

the cube. The same process is also used to determine the surface areas of prisms.

Similar to the cube, the area of a base of a rectangular prism can be calculated using the

area formula, B 5 3 w, where B is the area of the base, is the length of the rectangular

base, and w is the width of the rectangular base.

However, many of the prisms in this lesson do not have rectangular bases. The base of

a prism can be a variety of polygons. If the base of the prism is a pentagon, a hexagon,

an octagon, or any polygon different from a rectangle, you need to use a strategy to

determine the area of the base.

One base of a regular pentagonal prism is shown. Recall that when a polygon is regular,

that means all of the sides of the polygon are equal in length and all of the angles of the

polygon are equal in measure.

1. Locate and place a point at the center of the pentagon. From the center point, draw

line segments to connect the point with each vertex of the pentagon.

2. Describe the new polygons formed by adding these line segments.

3. What information do you need to calculate the area of each new polygon?

4. What formula is used to calculate the area of each new polygon?

5. Describe a strategy to determine the area of the entire pentagonal base.

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One base of a regular hexagonal prism is shown.

6. Use the same strategy you used in Question 1 to divide the hexagon into new polygons.

7. Describe the new polygons formed by adding these line segments.

8. What information do you need to calculate the area of each new polygon?

9. What formula is used to calculate the area of each new polygon?

10. Describe a strategy to determine the area of the entire hexagonal base.

11. Do you think this strategy works for any regular polygonal base of a prism? Explain

your reasoning.

In conclusion, the surface area of a prism is the sum of the areas of all of the lateral faces

of the prism plus the areas of the two bases.

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Problem 3 Volume Formula of a Prism

The formula for calculating the volume of a cube is V 5 B 3 h, where V represents the

volume of the cube, B is the area of the base of the cube, and h is the height of the cube.

The same is true for prisms.

Use the same formula, but apply it to a prism.

1. What does the variable V represent?

2. What does the variable B represent?

3. What does the variable h represent?

4. Write the formula for determining the volume of a prism. Define all variables used in

the formula.

5. Describe the strategy used for determining the area of the base of the prism when the

base is a regular polygon but not rectangular.

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Talk the Talk

Each numerical answer describes the volume or the surface area of a right rectangular

prism. Which is it? How do you know?

1. 13.44

2 cm

4.8 cm

1.4 cm

2. 33.64

1.6 cm

5.2 cm 1.25 cm

3. 42.5 m2

4. 50.8 m3

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Prisms are named by the shape of their bases.

5. Name the polygons that best describe the bases of each prism.

a. a pentagonal prism

b. an octagonal prism

c. a triangular prism

d. a decagonal prism

e. a hexagonal prism

f. a heptagonal prism

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6. Use the nets shown to determine the name of each prism.

a. b.

c. d.

e. f.


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14.5. . . Pyramids. . . •. . . 963

The Egyptians Were on to Something—or Was It the Mayans? Pyramids

The.Great.Pyramids.of.Egypt.are.a.favorite.tourist.spot.for.any.travelers.in.

the.area..Not.to.be.outdone,.the.pyramids.of.Mexico.are.also.a.favorite.tourist.

attraction.and.quite.challenging.to.climb!.Egypt.does.have.what.is.considered.to.

be.the.oldest.pyramid.in.the.world..The.Step.Pyramid.in.Saqqara.is.considered.to.

be.the.oldest.stone.pyramid..Some.experts.date.the.pyramid.was.built.between.

2649.and.2575.BC!.How.do.you.think.archaeologists.determine.the.age.of.a.

structure?.Do.you.think.there.was.a.reason.why.two.civilizations.were.alike.in.

building.pyramids?

Key Terms. pyramid

. vertex.of.a.pyramid

. height.of.a.pyramid

. slant.height.of.a.pyramid


. Create.a.model.of.a.pyramid.from.a.net.

. Use.nets.to.provide.two-dimensional.representations.

of.a.pyramid.

. Use.nets.to.estimate.the.surface.area.of.a.pyramid.

. Use.appropriate.units.of.measure.when.computing.

the.surface.area.of.a.pyramid.

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Problem 1 Getting to Know Pyramids

A pyramid is a polyhedron with one base and the same number of triangular faces as

there are sides of the base. The triangular faces are called lateral faces.

The vertex of a pyramid is the point at which all lateral

faces intersect.

All of the pyramids associated with this chapter have a

vertex that is located directly above the center point of

the base of the pyramid.

1. Sketch the first thing that comes to your mind when you

hear the word pyramid.

2. Does your sketch have one base? Does your sketch have the

same number of triangular faces as there are sides of the base?

3. Identify the vertex of the pyramid on your sketch.


pyramid? Explain how the sketches are the same or different.

Don't pyramids have other

vertices also?

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14.5. . . Pyramids. . . •. . . 965

Similar to a triangle, a height of a pyramid is the length of a line segment drawn from the

vertex of the pyramid to the base. This line segment is perpendicular to the base.

5. Use your sketch to explain what is meant by the “height of a pyramid.”

A slant height of a pyramid is the distance measured along

a lateral face from the base to the vertex of the pyramid

along the center of the face. As shown, a slant height, s, is

the altitude of a triangular lateral face of the pyramid.

S

The height of a triangular face of a pyramid is a dimension often needed to

calculate the total surface area of the pyramid.

Do you notice the

right angle symbol where the slant

height touches the base?

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Problem 2 Characteristics of a Pyramid

Use pasta and miniature marshmallows to construct a model of each pyramid. Use your

model to answer questions about the pyramids.

1. Construct and analyze this pyramid.

a. Name the polygon that is the base of this pyramid.

b. How many faces of the pyramid are lateral faces?

c. Describe the intersection of all of the lateral faces.

d. How many vertices, edges, and faces are in your model?

e. How can you determine the height of your pyramid?

Just like last time, you can break the pasta to any length

you want.

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14.5. . . Pyramids. . . •. . . 967





d. How many vertices, edges, and faces are there?


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e. How can you determine the height of this pyramid?

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14.5. . . Pyramids. . . •. . . 969







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5. Organize the data from Questions 1, 2, 3, and 4 by completing the table shown.

Shape of the Base of Pyramid

(Regular Polygon)

Number of Sidesof the Base

Number of Vertices

Number of Edges

Number of Faces

6. Use the table you completed in Question 5 to answer each question.

a. What is the relationship between the number of sides of the base and the number

of vertices of each pyramid?

b. What is the relationship between the number of sides of the base and the number

of edges of each pyramid?

c. What is the relationship between the number of sides of the base and the number

of faces of each pyramid?

7. Without making a model or drawing a sketch, predict the number of vertices, edges, and

faces for an octagonal pyramid. Describe your reasoning for making your prediction.

8. To verify your prediction, make a model of an octagonal pyramid to check your answers.

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14.5. . . Pyramids. . . •. . . 971

Problem 3 Pyramid Net

Mr. Morris instructed his math students to use their straw-and-marshmallow models of a

square pyramid to help them create a net.

Shawna raised her hand and said that she had an idea. She said that all she had to do

was remove the marshmallow that was at the top of the pyramid, lower the straws that

formed the lateral sides, and reuse the marshmallow somewhere else, but she would need

3 additional marshmallows to complete the net.

1. Sketch Shawna’s net of a square pyramid. Explain why she would need

3 additional marshmallows.

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2. Mr. Morris instructed his students to create a model of a pentagonal pyramid.

Then, he wanted them to create a net from their model. If Shawna uses the same

strategy she used to create the net for the square pyramid, how many additional

marshmallows will she need to build a net for a regular pentagonal pyramid?

3. Create Shawna’s net for a regular pentagonal pyramid.

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14.5 Pyramids • 973

4. Allen raised his hand and claimed that he created a different regular pentagonal

pyramid net. A drawing of Allen’s net is shown. How many additional marshmallows

will Allen need?

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14.5. . . Pyramids. . . •. . . 975

Problem 4 Surface Area of a Pyramid

Use Allen’s net of a regular pentagonal pyramid to estimate the surface area of a pyramid.

1. What information would you need to estimate the area of one triangle in Allen’s net?

2. Describe a strategy to estimate the area of the base of the pyramid in Allen’s net.

3. Use Allen’s net and a centimeter ruler to estimate the surface area of the pyramid.

Recall that the unit of measurement when estimating surface area is square units.

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4. Shawna is not convinced that Allen’s net is a pyramid because it looks different than

hers. Help to convince Shawna by cutting out, folding, and taping Allen’s net to show

it forms a regular pentagonal pyramid.

5. Do you think the strategy used to calculate the surface area of the regular pentagonal

pyramid also work for pyramids that have different regular polygonal bases?

You have just used various strategies to calculate surface area. However at this point, you

will not generate a formula for determining the surface area of a pyramid. You will explore

the surface area of a pyramid in depth when you study geometry in high school.


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Talk the Talk

1. Solve for the surface area of each pyramid.

a. A square pyramid where the length of each side of the base is 10 inches and the

slant height is also 10 inches.

b. The base of the pyramid is a regular pentagon.

6 cm

14 cm

3.5 cm

Like prisms, pyramids are named by the shape of their bases.

2. Name the polygon that best describes the base of each pyramid.

a. a pentagonal pyramid b. an octagonal pyramid

c. a triangular pyramid d. a decagonal pyramid

e. a hexagonal pyramid f. a heptagonal pyramid

14.5. . . Pyramids. . . •. . . 977


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3. Use the nets to determine the name of each pyramid.

a. b.

c. d.

e. f.


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. Identify.geometric.solids.

. Compare.and.contrast.the.surface.area.of.geometric.solids.

. Apply.the.surface.area.concept.to.a.real-world.situation.

And The Winning Prototype Is . . . ?Identifying Geometric Solids in Everyday Occurrences

When.was.the.last.time.you.saw.a.circle?.Or.perhaps,.when.was.the.last.time.

you.saw.a.line—in.the.geometric.terms?.In.fact,.when.you.begin.to.formally.study.

geometry.in.high.school,.most.of.your.instruction.will.begin.with.two-dimensional.

figures;.however,.the.world.is.full.of.three-dimensional.objects..Even.a.piece.of.

paper.may.“appear”.to.be.a.two-dimensional.object,.but.it.isn’t!.It.does.have.a.

depth,.even.though.that.depth.is.quite.small..

Why.do.you.think.that.most.geometry.courses.start.with.two-dimensional.

examples?.Do.you.think.there.are.some.principles.that.are.key.in.two-dimensional.

examples.that.will.be.used.when.studying.three-dimensional.objects?

14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 979

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Problem 1 Geometric Solids are Everywhere!

1. Geometric solids appear in real life in a variety of places. Identify each solid.

a. b.

c. d.

e. f.

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g. h.

i. j.

k. l.

m. n.

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Problem 2 Gathering Information

Throughout this chapter, you have estimated and calculated the volume and surface area

of four prototypes developed by the Rice Is Nice product development team.

It is now time to compile this information and develop a business plan to market each

prototype. The amount of money it costs the manufacturers to package a product and the

amount of money generated by the sale of this product will determine the profit margin.

1. Complete the table with the information of each prototype for Rice Is Nice.

Prototype Number

Name of the Geometric Solid

Surface Area (in cm2)

Prototype.#1

Prototype.#2

Prototype.#3

Prototype.#4

The nets representing each

of the prototypes can be found at the end of

the first lesson in this chapter.

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2. Match each sketch with the appropriate prototype number.

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3. Consumers are concerned about how much of the

product they get for their money. If the cost of each

rice container is the same, does the measurement

of the surface area help them to determine the

best buy? Explain.

4. Manufacturers are concerned with maximizing their profit. Does the

measurement of the surface area help them to determine the best

choice? Explain.

Problem 3 Commemorative Canisters

The product development team members came up with a great idea to introduce their new

rice container. They decided to give consumers a complimentary commemorative metal

canister with their first purchase of the newly packaged product. The metal canister will

maintain the same size and same shape as the prototype container.

The stockholders of the Rice Is Nice Company asked the development team to calculate

the cost of materials used to make the commemorative canisters for the four prototypes.

The team wants to compare the price of using aluminum, tin, and copper.

1. Calculate the cost of using aluminum. One rectangular sheet of aluminum measuring

25.4 centimeters long, 10.2 centimeters wide, and 0.04 centimeters thick will

cost $2.69.

a. How many square centimeters are in one rectangular sheet of aluminum?

b. Determine the cost of aluminum per square centimeter to the nearest tenth of

a cent. When calculating an amount of money, always round up.

How are surface area and volume related?

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c. Use the information from Problem 2 to complete the table.

Prototype Number


Surface Area (cm2)

Cost of Aluminum (dollars)

Prototype.#1

Prototype.#2

Prototype.#3

Prototype.#4

2. Calculate the cost of using tin. One rectangular sheet of tin measuring

25.4 centimeters long, 10.2 centimeters wide, and 0.02 centimeters thick will

cost $3.09.

a. How many square centimeters are in one rectangular sheet of tin?

b. Determine the cost of tin per square centimeter to the nearest tenth of a cent.

When calculating an amount of money, always round up.


Prototype Number


Surface Area (cm2)

Cost of Tin (dollars)

Prototype.#1

Prototype.#2

Prototype.#3

Prototype.#4

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3. Calculate the cost of using copper. One rectangular sheet of copper measuring

25.4 cm long, 10.2 cm wide, and 0.06 cm thick will cost $9.49

a. How many square centimeters are in one rectangular sheet of copper?

b. Determine the cost of copper per square centimeter to the nearest tenth of a cent.

When calculating an amount of money, always round up.


Prototype Number


Surface Area (cm2)

Cost of Copper (dollars)

Prototype.#1

Prototype.#2

Prototype.#3

Prototype.#4

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Talk the Talk

Team up with a few classmates to write a report to the director of marketing. In the report,

recommend one of the three prototypes for production, and the material that should be

used to produce the commemorative canister. Explain your reasoning. Then, present

your report to the class and the class can decide which report is the most convincing.



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Constructing a Net from a Model of a Geometric Solid

Geometric solids are bounded three-dimensional geometric

figures. The three dimensions are length, width, and

height. A net is a two-dimensional representation

of a geometric solid. A net can be cut out,

folded, and glued or taped to create a model of

a geometric solid. When constructing a net, it

may be helpful to make a sketch of each side or

face of the geometric solid first. Then, connect

each side in such a way that they share common

edges. When folded along these edges, the net

should be a model of the geometric solid.

Example

A net is sketched from the given model of a geometric solid.

Key Terms geometricsolids(14.1)

prototype(14.1)

edge(14.1)

face(14.1)

vertex(14.1)

net(14.1)

point(14.2)

linesegment(14.2)

polygon(14.2)

polyhedron(14.2)

regularpolyhedron(14.2)

congruent(14.2)

cube(14.2)

unitcube(14.2)

diameter(14.2)

surfacearea(14.2)

volume(14.2)

prism(14.3)

basesofaprism(14.3)

lateralfaces(14.3)

heightofaprism(14.3)

rectangularprism(14.3)

rightprism(14.3)

pyramid(14.5)

vertexofapyramid(14.5)

heightofapyramid(14.5)

slantheightofapyramid(14.5)

Chapter 14 Summary

Chapter 14 Summary • 989

Whoo! That was a tough

chapter and I know I made a lot of mistakes but you know, as Einstein said,

"A person who never made a mistake, never tried

anything new."


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Notice how the faces that share sides in the net share common edges in the model of the

cube. The other faces that share edges in the model are connected by the tabs.

top

left front right back

bottom

Calculating the Surface Area and Volume of Cubes

A polyhedron is a 3-dimensional solid that has polygons as faces. A regular polyhedron

has congruent regular polygons as faces and has congruent angles between all faces. A

cube is a regular polyhedron whose six faces are congruent squares. Surface area is the

total area of the 2-dimensional surfaces that make up a 3-dimensional object. The surface

area of a cube is calculated by determining the area of one face and then multiplying

that area by 6. Volume is the amount of space occupied by an object. To calculate the

volume of a cube, use the formula V 5 B 3 h, where V represents the volume of the cube,

B represents the area of the base of the cube, and h represents the height of the cube.

Example

7 cm

To determine the area of one face of the cube, multiply the length times the width.

Area of the base: 7 3 7 5 49 cm2

Surface area of the cube: 6 3 49 5 294 cm2

Use the formula V 5 B 3 h to calculate the volume of the cube.

V 5 B 3 h

5 49 3 7

5 343

The volume of the cube is 343 cm3.

When a cube’s dimensions are doubled, the volume of the resulting cube is eight times the

volume of the initial cube. If the given cube’s dimensions were doubled, the volume would

be 8 3343 52744 cm3.

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Calculating the Surface Area and Volume of Right Rectangular Prisms

A prism is a polyhedron with two parallel and congruent faces called bases. All other faces

are parallelograms and are referred to as lateral faces. A rectangular prism is a prism that

has rectangles as its bases. A right prism is a prism that has bases aligned one directly

above the other and has lateral faces that are rectangles. To calculate the surface area of a

right rectangular prism, calculate the area of each rectangular face and add each of these

areas together. As with cubes, the volume of a prism can be determined by using the

formula V 5 B 3 h, where V represents the volume of the prism, B represents the area of

the base of the prism, and h represents the height of the prism.

Example

7 cm

3 cm

2 cm

Surface area of the right rectangular prism:

2(7 32) 1 2(7 33) 1 2(2 33)

5 28 1 42 1 12

5 82 cm2

Area of the base of the prism: 7 32 5 14 cm2

Use the formula V 5 B 3 h to calculate the volume of the prism.

V 5 B 3 h

5 14 3 3

5 42

The volume of the prism is 42 cm3.

Chapter 14. . . Summary. . . •. . . 991


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Calculating the Surface Area and Volume of Prisms

A prism is a polyhedron with two parallel and congruent faces called bases. All other faces

are parallelograms and are referred to as lateral faces. The base of a prism can be any of a

variety of polygons. To calculate the surface area of a prism that does not have a rectangle

as a base, calculate the area of each face and add the areas together. The volume of a

prism can be determined by using the formula V 5 B 3 h, where V represents the volume

of the prism, B represents the area of the base of the prism, and h represents the height of

the prism.

Example

4.2 cm

5 cm

4.2 cm

5 cm

base

6 cm

5 cm

lateralface

To determine the area of the base, divide it into six congruent triangles and multiply the

area of one triangle by 6. Recall that the area formula for a triangle is A 5 1 __ 2

3b 3h,

where A represents the area of the triangle, b represents the length of the triangle’s base,

and h represents the height of the triangle.

Area of one triangular piece of the base: 1 __ 2

35 34.2 510.5 cm2

Area of the base of the prism: 6 310.5 563 cm2

Area of each lateral face: 635 530 cm2

Surface area of prism 5 Area of base 1 Area of lateral faces

5 63 1 (6 3 30)

5 63 1 180

5 243

The surface area of the prism is 243 cm2.

Use the formula V 5 B 3 h to calculate the volume of the prism.

V 5 B 3 h

5 243 3 6

5 1458 cm3

The volume of the prism is 1458 cm3.

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Calculating the Surface Area of Pyramids

A pyramid is a polyhedron with one base. The other faces of the pyramid are called lateral

faces and each lateral face is in the shape of a triangle. The number of lateral faces is

equal to the number of sides of the base. The vertex of a pyramid is the point at which all

lateral faces intersect. All of the pyramids associated with this chapter have a vertex that

is located directly above the center point of the base of the pyramid. The surface area is

determined by adding the areas of the base and each lateral face.

Example

8 cm

base

lateralface

13.2 cm

5.6 cm

To determine the area of the base, divide it into five congruent triangles and multiply the

area of one triangle by 5. Recall that the area formula for a triangle is A 5 1 __ 2

3b 3h,

where A represents the area of the triangle, b represents the length of the triangle’s base,

and h represents the height of the triangle.

Area of one triangular piece of the base: 1 __ 2

38 35.6 522.4 cm2

Area of the base of the pyramid: 5 322.4 5112 cm2

Area of each lateral face: 1 __ 2

38 313.2 552.8 cm2

Surface area of pyramid 5 Area of base 1 Area of lateral faces

5 112 1 (5 3 52.8)

5 112 1 264

5 376

The surface area of the pyramid is 376 cm2.

Chapter 14. . . Summary. . . •. . . 993


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Applying Surface Area Concepts to a Real World Situation

Many objects in the real world are geometric solids. People in a variety of jobs must apply

the surface area concepts studied in this chapter in order to solve real world problems

involving geometric solids.

Example

Horatio plans to construct steel grain bins in the shape of right rectangular prisms. The

steel sheeting he will use costs $3.50 per square foot. The top face of the grain bin will be

attached with three hinges to allow it to open and close. The cost of each hinge is $1.50.

How much will the material for one grain bin cost?

10 feet

3.5 feet

4 feet

Surface area of right rectangular prism: 2(10 3 3.5) 1 2(10 3 4) 1 2(4 3 3.5) 5 70 1 80 1

28 5 178 ft2

The cost of the steel sheeting for one grain bin is 3.50(178) 5 $623.00.

The cost of the hinges for one grain bin is 1.50(3) 5 $4.50.

The total material cost for one grain bin is $623.00 1 $4.50 5 $627.50.

Documents

Introduction to the Third Dimensionmrpack.weebly.com/uploads/8/5/0/5/8505687/course_1_chapter_14.… · Sketch.various.views.of.a.solid.figure.to.provide.a. two-dimensional.representation.of.a.three-dimensional.figure