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Math 8

Chapter 7

Name _________

Page Values: Lesson Reading Guide: Level 1 Study Guide and Intervention: Level 1 Skills Practice: Level 1 Practice: Level 2 Word Problem: Level 2 Enrichment: Level 3

Vocabulary TermFound

Definition/Description/Exampleon Page

base

center

circumference

chord

complex figure

cone

cylinder

diameter

edge

face

lateral face

lateral surface area

Chapter 7 1 Course 3

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7

NAME ________________________________________ DATE ______________ PERIOD _____

Student-Built Glossary

This is an alphabetical list of new vocabulary terms you will learn inChapter 7. As you study the chapter, complete each term’s definitionor description. Remember to add the page number where you foundthe term. Add this page to your math study notebook to reviewvocabulary at the end of the chapter.

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NAME ________________________________________ DATE ______________ PERIOD _____

Student-Built Glossary (continued)

Vocabulary TermFound

Definition/Description/Exampleon Page

net

pi

plane

polyedron

prism

pyramid

radius

regular pyramid

similar solids

slant height

total surface area

vertex

volume


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7

NAME ________________________________________ DATE ______________ PERIOD _____

Family Letter

Dear Parent or Guardian:

Learning about geometry is exciting because geometric shapes

are everywhere! From the food we eat to the things we build, we

are influenced by geometric shapes. Knowing how to find the

areas and volumes of these shapes helps us make decisions such

as the amount of material we need to construct an object or the

amount of liquid we need to fill a container. These types of

decisions are made in almost every industry.

In Chapter 7, Measurement: Area and Volume, your child

will learn how to calculate circumference and area of circles, the

area of composite figures, to find surface areas and volumes of

prisms, cylinders, pyramids, and cones. Your child will also learn

about similar figures and to solve problems by solving a simpler

problem. In the study of this chapter, your child will complete a

variety of daily classroom assignments and activities and

possibly produce a chapter project.

By signing this letter and returning it with your child, you

agree to encourage your child by getting involved. Enclosed is

an activity you can do with your child that practices how the

math we will be learning in Chapter 7 might be tested. You

may also wish to log on to glencoe.com for self-check quizzes

and other study help. If you have any questions or comments,

feel free to contact me at school.

Sincerely,

Signature of Parent or Guardian ______________________________________ Date ________

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NAME ________________________________________ DATE ______________ PERIOD _____

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Anticipation Guide

Measurement: Area and Volume

Before you begin Chapter 7

N Read each statement.

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

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

STEP 1 STEP 2

A, D, or NSStatement

A or D

1. The distance from the center of a circle to any point on the circle

is called the radius.

2. The diameter of a circle equals two times the radius.

3. The formula for the area of a circle is A 5 2πr or πd.

4. The area of a composite figure can be found by separating it into

shapes whose areas you know how to find.

5. A rectangular prism has six edges, six faces, and eight vertices.

6. A rectangular pyramid has a rectangular base and four

triangular faces.

7. Measurements of volume are given in cubic units.

8. The volume of any prism can be found by the formula V 5 lwh.

9. The volume of a rectangular prism with the same base and

height as a rectangular pyramid will be }1

3} that of the pyramid.

10. The surface area of three-dimensional solids is given in square

units.

11. The height and slant height of a pyramid are the same.

12. If two rectangular prisms are similar with a scale factor of 2,

then the volume of the larger prism will be 6 times the volume

of the smaller prism.

After you complete Chapter 7

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

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

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

Step 1

Step 2

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NAME ________________________________________ DATE ______________ PERIOD _____

Lesson Reading Guide

Circumference and Area of Circles

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Get Ready for the Lesson

Complete the Mini Lab at the top of page 352 in your textbook.

Write your answers below.

1. What distance does C represent?

2. Find the ratio }Cd

} for this object.

3. Repeat the steps above for at least two other circular objects and compare

the ratios of C to d. What do you observe?

4. Graph the data you collected as ordered pairs, (d, C). Then describe the

graph.

Read the Lesson

5. Explain the difference between the radius and the diameter of a circle.

6. What is the ratio of the circumference of a circle to its diameter?

7. Explain how you find the circumference of a circle given its radius is

4 inches.

Remember What You Learned

8. One way to help you remember a formula or concept is to make up a

saying. For example, to remember the formula for the area of a circle you

might use, “Fuzzy Wuzzy was a bear; area equals p (pi) r squared.” Make

up your own sayings to help you remember the formulas for the

circumference and area of circles.

C

d

O 1 2 3 4 5

3

6

9

12

Diameter (cm)

Cir

cum

fere

nce

(cm

)


NAME ________________________________________ DATE ______________ PERIOD _____

Study Guide and Intervention


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Exercises

Examples

Example 3

Find the circumference of each circle. Use 3.14 for p. Round to thenearest tenth.

C 5 pd Circumference of a circle

C 5 p · 4 Replace d with 4.

C 5 4p This is the exact circumference.

C < 4 · 3.14 or 12.6 Replace p with 3.14 and multiply.

The circumference is about 12.6 inches.

C 5 2pr Circumference of a circle

C < 2 · 3.14 · 5.4 Replace r with 5.4.

C < 33.9 Replace p with 3.14 and multiply.

The circumference is about 33.9 meters.

Find the area of the circle. Use 3.14 for p. Round to the nearesttenth.

A 5 pr2 Area of a circle

A < 3.14(1.5)2 Replace p with 3.14 and r with half of 3 or 1.5.

A < 3.14 · 2.25 Evaluate (1.5)2.

A < 7.1 Multiply.

The area is about 7.1 square feet.

Find the circumference and area of each circle. Use 3.14 for p. Round to thenearest tenth.

1. 2. 3.

4. The diameter is 9.3 meters.

5. The radius is 6.9 millimeter.

6. The diameter is 15.7 inches.

4.2 m

11 yd

1 cm

3 ft

5.4 m

4 in.

The circumference C of a circle is equal to its diameter d times p or 2 times the radius r times p, or C 5 pd or C 5 2pr.

The area A of a circle is equal to p times the square of the radius r, or A 5 pr 2.

r d

C


Skills Practice


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Find the circumference and area of each circle. Use 3.14 for p. Roundto the nearest tenth.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. The diameter is 7.7 feet. 11. The radius is 9.6 millimeters.

12. The radius is 3.8 meters. 13. The diameter is 17.4 yards.

14. The radius is 11.3 centimeters. 15. The diameter is 4}34

} miles.

16. The radius is 2}13

} inches. 17. The diameter is 7}58

} feet.

18. The radius is 5.25 meters. 19. The diameter is 12}34

} yards.

6 m34

2 ft45 11.6 km

8.3 mi

5.7 mm

1.9 yd

12 in. 4 m

1 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Practice


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Find the circumference of each circle. Use 3.14 for p. Round to the nearest tenth.

1. 2. 3. 4.

Find the area of each circle. Use 3.14 for p. Round to the nearest tenth.

5. 6. 7. 8.

Find the circumference and area of each circle. Round to the nearest tenth.

9. The diameter is 8 centimeters. 10. The radius is 4.7 inches.

11. The radius is 0.9 feet. 12. The diameter is 6.8 kilometers.

Another approximate value for π is }2

7

2}. Use this value to find the circumference

and area of each circle.

13. The diameter is 14 yards. 14. The radius is 1}1

6} millimeters.

15. WINDMILL Each sail on a windmill is 5 meters in length.

How much area do the wings cover as they turn from the

force of the wind?

16. ALGEBRA Find the radius of a circle if its area is

314 square miles.

10 in.

14 mm

22 yd 25 m

25 m

8.5 ft

6.75 mi

5.25 cm


Word Problem Practice


NAME ________________________________________ DATE ______________ PERIOD _____

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3. GARDENING A flowerpot has a circular

base with a diameter of 27 centimeters.

Find the circumference of the base of

the flowerpot. Round to the nearest

tenth.

4. WINDOWS Find the area of the window

shown below. Round to the nearest

tenth.

36 in.

1. FOUNTAINS The circular fountain in

front of the courthouse has a radius of

9.4 feet. What is the circumference of

the fountain? Round to the nearest

tenth.

2. PETS A dog is leashed to a point in the

center of a large yard, so the area

the dog is able to explore is circular.

The leash is 20 feet long. What is the

area of the region the dog is able to

explore? Round to the nearest tenth.

5. BICYCLES A bicycle tire has a radius of

13}14

} inches. How far will the bicycle

travel in 40 rotations of the tire? Round

to the nearest tenth.

13 in.14

6. LANDSCAPING Joni has a circular

garden with a diameter of 14}12

} feet. If

she uses 2 teaspoons of fertilizer for

every 25 square feet of garden, how

much fertilizer will Joni need for her

entire garden? Round to the nearest

tenth.


NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment


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Angles and Arcs

A central angle is an angle that intersects a circle in two points and has its vertex at the center of thecircle. It separates a circle into a major arc and aminor arc.

• The degree measure of a minor arc is the degree

measure of the central angle. In circle B,

mACC

5 m/ABC.

• The degree measure of a major arc is 360 minus

the degree measure of the central angle. In circle B,

mADCC

5 3608 2 m/ABC.

An inscribed angle has its vertex on the circle and sides that contain chords. The measure of an inscribed angle equals one-half the measure of its intercepted arc. In the circle shown at the right,

m/XYZ 5 }12

}mXZC

. Thus, mXZC

5 2 ? m/XYZ.

Find the measure of each arc.

1. 2. 3.

minor arc LN minor arc QS major arc VT

Refer to the diagram at the right. Find the measure of each of the following angles or arcs.

4. minor arc JG 5. /1

6. major arc GJ 7. /2

8. minor arc KH 9. minor arc GK

10. minor arc FH 11. /FJK

12. /JFG 13. arc HJG

F

K G

H

12 1108

J

258

T

V

UQ

S

R

84˚M

L

N

115˚

X

Y

Z

inscribedangle

XYZ

central angle

ABC

BD

A

C

minor arc AC

major arc CA


NAME ________________________________________ DATE ______________ PERIOD _____

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Example 1

Example 2

Exercises

Gift cards come in packages of 12 and envelopes come in packagesof 15. Meagan needs to send 600 cards in envelopes. How manypackages of each kind should she buy?

Understand Meagan needs that same number of cards and envelopes.

Plan Find out how many packages are needed for 300 cards in envelopes.

Solve 12c 5 300 15e 5 300

c 5 25 e 5 20

Multiply the answers by 2.

Check 2 3 25 5 50 packages of cards 2 3 20 5 40 packages of envelopes

Meagan should buy 50 packages of cards and 40 packages of envelopes.

How many triangles of any size are in the figure at the right?

Understand We need to find how many triangles are in the figure.

Plan Draw a simpler diagram.

Solve 9 Count the smallest triangles, which have 1 triangle per side.

3 Count the next largest triangles, which have 2 triangles per side.

1 Count the largest triangle, which has 3 triangles per side.

13 Add together to find the total triangles of any size.

Check Now repeat the steps for the original problem.

16 Count the smallest triangles, which have 1 triangle per side.



1 Count the largest triangle, which has 4 triangles per side.

27 Add together to find the total triangles of any size.

For Exercises 1–3, solve a simpler problem.

1. Hot dogs come in packages of 10 and buns come in packages of 8. How many packages

of each will Mindy need to provide 640 hot dogs for a street fair?

2. Mark can plant 3 tree saplings in an hour and Randy can plant 5 tree saplings in an

hour. Working together, how long will it take them to plant 80 tree saplings?

3. A restaurant has 18 square tables that can be pushed together to form one long table

for large parties. Each square table can seat 2 people per side. How many people can be

seated at the combined tables?


Problem-Solving Investigation: Solve a Simpler Problem


For Exercises 1–3, rewrite the problem as a simpler problem.

1. Jerry has a square-shaped deep-dish pizza. What is the maximum number of pieces

that can be made by using 6 cuts?

2. CDs come in packages of 25 and CD cases come in packages of 16. How many of each

type of package will Lilly need to buy in order to make print 400 CDs and put them in

cases with none left of either?

3. A restaurant has 10 triangular tables that can be pushed together in an alternating up-

and-down pattern as shown below to form one long table for large parties. Each

triangular table can seat 3 people per side. How many people can be seated at the

combined tables?

For Exercises 4–15, rewrite to solve a simpler problem and solve.Find a reasonable answer.

4. 13 3 29 5. 48 1 32 1 87

6. 74 3 (18 2 9) 7. 33 4 9

8. }1

5

1

7

3} 9. 55 1 44 1 33

10. 63 3 17 11. 532 2 389

12. 78 3 41 2 276 13. 52 1 39 1 111

14. 452 2 377 15. 67 3 34 3 12

NAME ________________________________________ DATE ______________ PERIOD _____

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Skills Practice



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Use the solve a simpler problemstrategy to solve Exercises 1 and 2.

1. ASSEMBLY A computer company hastwo locations that assemble computers.One location assembles 13 computers inan hour and the other locationassembles 12 computers in an hour.Working together, how long will it takeboth locations to assemble 80computers?

2. AREA Determine the area of theshaded region if the radii of the sixcircles are 1, 2, 3, 4, 5, and 10centimeters. Use 3.14 for p. Round tothe nearest tenth if necessary.

Use any strategy to solve Exercises 3–6.Some strategies are shown below.

3. NUMBER SENSE Find the sum of all theeven numbers from 2 to 50, inclusive.

4. ANALYZE TABLES Mr. Brown has $1,050to spend on computer equipment. DoesMr. Brown have enough money to buythe computer, scanner, and software if a20% discount is given and the sales taxis 5%? Explain.

5. COPIER The counter on a businesscopier read 18,678 at the beginning ofthe week and read 20,438 at the end ofthe week. If the business was inoperation 40 hours that week, what wasthe average number of copies made eachhour?

6. HUMMINGBIRD In normal flight ahummingbird can flap its wings 75times each second. At this rate, howmany times does a hummingbird flap itwings in a 20-minute flight?

PROBLEM-SOLVING STRATEGIES

N Look for a pattern.

N Use a Venn diagram.

N Solve a simpler problem.

Item Cost

Computer $899

Scanner $54

Software $278

Mixed Problem Solving

Practice


For Exercises 1–6, use the solve a simpler problem strategy.


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3. PACKAGES Postcards come in packages

of 12 and stamps come in packages of

20. How many of each type of package

will Jessica need to buy in order to

send 300 postcards with no stamps or

postcards left over?

4. JOBS Larry can stuff 150 envelopes in

one hour. Harold can stuff 225

envelopes in one hour. About how long

will it take them to stuff 10,000

envelopes?

1. GEOMETRY Mark has a large pizza.

What is the maximum number of

pieces that can be made by using 12

cuts?

2. TABLES A picnic area has 21 square

tables that can be pushed together to

form one long table for large group.

Each square table can seat 4 people per

side. How many people can be seated at

the combined tables?

5. BUILDING Jason can lay 40 bricks in

one hour. Mark can lay 30 bricks in one

hour. Jesse can lay 20 bricks in one

hour. About how long will it them to

build a wall that uses 900 bricks?

6. GEOMETRY How many squares of any

size are in the figure?




NAME ________________________________________ DATE ______________ PERIOD _____

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Area of Composite Figures


Read the introduction at the top of page 363 in your textbook.


1. Identify some of the polygons that make up the infield of the speedway.

2. How can the polygons be used to find the total area of the infield?

Read the Lesson

3. What is a composite figure?

4. What is the first step in finding the area of a composite figure?

5. Explain how to divide up the figure shown.


6. Look up the everyday definition of the word composite in a dictionary.

How does the definition relate to what you learned in the lesson?




NAME ________________________________________ DATE ______________ PERIOD _____

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Find the area of the composite figure.

The figure can be separated into a semicircle and trapezoid.

Area of semicircle Area of trapezoid

A 5 }12

}pr2 A 5 }12

}h(b1 1 b2)

A 5 }12

} ? 3.14 ? (7)2 A 5 }12

} ? 10 ? (14 1 18)

A < 77.0 A 5 160

The area of the figure is about 77.0 1 160 or 237 square inches.

Find the area of each figure. Use 3.14 for p. Round to the nearest tenth ifnecessary.

1. 2. 3.

4. What is the area of a figure formed using a triangle with a base of

6 meters and a height of 11 meters and a parallelogram with a base of

6 meters and a height of 11 meters?

5. What is the area of a figure formed using a semicircle with a diameter of

8 yards and a square with sides of a length of 6 yards?

6. What is the area of a figure formed using a rectangle with a length of

9 inches and a width of 3 inches and a triangle with a base of 4 inches

and a height of 13 inches?

7 mi

7 mi

5 mi

5 mi

14 mi

6 ft

9 ft

9 ft

5 mm

8 mm

6 mm

10 in.

14 in.

18 in.

To find the area of a composite figure, separate the figure into shapes whose areas you know how tofind. Then find the sum of these areas.

Exercises

Example


NAME ________________________________________ DATE ______________ PERIOD _____

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Skills Practice

Area of Composite FiguresFind the area of each figure. Use 3.14 for p. Round to the nearesttenth if necessary.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. What is the area of a figure formed using a semicircle with a diameter of

16 feet and a trapezoid with a height of 8 feet and bases of 12 feet and

14 feet?

11. What is the area of a figure formed using a rectangle with a length of

13 kilometers and a width of 7 kilometers and a triangle with a base of

14 kilometers and a height of 11 kilometers?

12 km

4 km

4 km

5 km12 m

10 m

13 m13 m7 m

17 m

7 m

14 m

8 m

14 m 6 m6 m

5 in.

9 in.

4 in.

8 in.

4 in. 6 in.

10 in.

18 in.

6 in.

6 cm

5 cm

5 ft3 ft

6 ft

4 ft

5 cm

7 cm

10 cm

14 cm

6 cm

12 yd

12 yd

10 m

7 m6 m


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Practice


Find the area of each figure. Use 3.14 for p. Round to the nearest tenth ifnecessary.

1. 2. 3.

4. 5. 6.

In each diagram, one square unit represents 10 square centimeters.Find the area of each figure. Round to the nearest tenth if necessary.

7. 8.

9. GAZEBO The Parks and Recreation department

is building a gazebo in the local park with the

dimensions shown in the figure. What is the area

of the floor?

10. DECK The Pueyo family wants to paint the deck

around their swimming pool with the dimensions

shown in the figure. If a gallon covers 200 square

feet, how many gallons of paint are needed to apply

two coats of paint?

5 mi

12 mi

8 mi18 mi

5.9 cm

3.6 cm

1.1 cm

4.8 cm

6 m8 m

10 m 6 m

20 m

24 ft

36 ft24 ft

12 ft

30 ft

18 ft

5 m

11 m

4 m

8 yd

9 yd

4 in.

12 in.

7 in.

9 in.

5 ft

4 ft


NAME ________________________________________ DATE ______________ PERIOD _____



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LANDSCAPING For Exercises 1 and 2 use the diagram of a yard and the following information. The figure shows the measurements of Marcus’ yard which he intends to sod.

15 ft

20 ft

30 ft

50 ft

3. ICE CREAM Leeor was asked to repaint

the sign for his mother’s ice cream

shop, so he needs to figure out how

much paint he will need. Find the area

of the ice cream cone on the sign.

Round to the nearest tenth.

6 in.

12 in.

4. HOME IMPROVEMENT Jim is planning to

install a new countertop in his kitchen,

as shown in the figure. Find the area of

the countertop.

3 ft

6 ft

3 ft

2.5 ft

2 ft

3 ft 2 ft2 ft

2.5 ft

1. Find the area of the yard. 2. One pallet of sod covers 400 square

feet. How many full pallets of sod will

Marcus need to buy to have enough for

his entire yard?

5. SCHOOL PRIDE Cindy has a jacket with

the first letter of her school’s name on

it. Find the area of the letter on Cindy’s

jacket.

2 in.

10 in.2 in.

2 in.

6 in.

6 in.

6. SWIMMING POOLS The Cruz family is

buying a custom-made cover for their

swimming pool, shown below. The cover

costs $2.95 per square foot. How much

will the cover cost? Round to the

nearest cent.

15 ft

25 ft


Enrichment

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Inuit Architecture

The Inuit are a Native American people who live primarily in the arctic regions of Alaska,Canada, Siberia, and Greenland. The Inuit word iglu means “winter house,” and it originally referred to any permanent structure used for shelter in the winter months. In the nineteenth century, however, the term came to mean a domed structure built of snow blocks, as shown in the figure at the right.

An iglu could shelter a family of five or six people. Sometimes several families built a cluster of iglus that were connected by passageways and shared storage and recreation chambers. The figure below is a drawing of such a cluster. Use the drawing to answer each of the following questions. When appropriate, round answers to the nearest whole number.

1. What is the circumference

of the entry chamber?

2. What is the circumference of

one of the living chambers?

3. Estimate the distance from

the front of the entry

chamber to the back of

the storage chamber.

4. An iglu is a hemisphere, or

half a sphere. The formula

for the volume of a sphere is

V 5 }43

}pr3, where r is the

radius. Estimate the volume

of the storage chamber.

STORAGE

LIVINGLIVING

LIVING LIVING

ENTRY

RECREATION

8 ft

8 ft8 ft

12 ft

6 ft

8 ft

10 ft


NAME ________________________________________ DATE ______________ PERIOD _____


Three-Dimensional Figures

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1. Name the two-dimensional shapes that make up the sides of the building.

2. If you observed the building from directly above, what two-dimensional

figure would you see?

3. How are two- and three-dimensional figures related?

Read the Lesson

4. A plane is a two-dimensional flat surface that extends in all directions.

What is formed when two planes intersect?

5. How can you tell the difference between a prism and pyramid?

6. Identify a three-dimensional figure that has one base that is a hexagon

and six other faces that are triangles.


7. Visualize your classroom as a prism with yourself sitting in the middle of

the room. What parts of the classroom represent an edge, a face, and a

vertex?




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Identify the solid. Name the number and shapesof the faces. Then name the number of edges and vertices.

The figure has one base that is a triangle, so it is a triangular pyramid.The other faces are also triangles. It has a total of 4 faces, 6 edges, and 4 vertices.

Identify each solid. Name the number and shapes of the faces. Thenname the number of edges and vertices.

1. 2. 3.

4. Draw and label the top, front, and side views of the

chair shown.

face

face

edge

vertex

A polyhedron is a three-dimensional figure with flat surfaces that are polygons. A prism is apolyhedron with two parallel, congruent faces called bases. A pyramid is a polyhedron with one basethat is a polygon and faces that are triangles. Prisms and pyramids are named by the shape of theirbases.

Exercises

Example


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Skills Practice


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1. 2. 3.

4. 5. 6.

7. 8. 9.


Practice


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1. 2. 3.

4. Name a plane that is parallel to plane ABCD.

5. Identify a segment that is skew to segment YZ.

Identify each solid.

6.

7.

A

B C

Y

ZW

D

X


NAME ________________________________________ DATE ______________ PERIOD _____



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ARCHITECTURE For Exercises 1–3, refer to the architectural drawing of a table.

FrontSide

3. Find the area of the shaded region. 4. NAVIGATION Sailing ships once used

deck prisms to allow sunlight to reach

below the main deck. One such deck

prism is shown below. Identify the

solid. Name the number and shapes of

the faces. Then name the number of

edges and vertices.

1. Draw and label the top, front, and side

views of the table.

2. Find the overall height of the table in

feet.

5. PUBLIC SPEAKING A pedestal used in an

auditorium is shaped like a rectangular

prism that is 1 unit high, 5 units wide,

and 5 units long. Sketch the pedestal

using isometric dot paper.

6. PETS Lisa has four pet fish that she

keeps in an aquarium. The aquarium is

shaped like a triangular prism that is

4 units high. Sketch what this

aquarium might look like using

isometric dot paper.


Enrichment

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The Five Platonic Solids

There are only five regular convex solids. They are called the Platonic Solidsand are shown here.

1. Write the name of each Platonic Solid under its net.

2. Complete this chart for the Platonic Solids.

3. Write an equation relating the number of faces, edges, and vertices of

the Platonic Solids. This equation is called Euler’s Formula and is true

for all simple polyhedra.

tetrahedron hexahedron octahedron icosahedron dodecahedron

Solid Tetrahedron Hexahedron Octahedron Icosahedron Dodecahedron

Number of Faces

Number of Edges

Number of Vertices


NAME ________________________________________ DATE ______________ PERIOD _____


Volume of Prisms and Cylinders

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1. Describe how the volume V of each prism is related to its length ,,

width w, and height h.

2. Describe how the area of the base B and the height h of each prism is

related to its volume V.

Read the Lesson

3. What is another way to write the volume of a rectangular prism other

than V 5 Bh?

4. What does it mean if a figure has a volume of 120 cubic centimeters?

5. Explain how finding the volume of a composite solid is similar to finding

the area of a composite figure.


6. Complete the table below by filling in the correct formula.

Figure Formula for Finding Volume

rectangular prism

triangular prism

cylinder




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Find the volume of the rectangular prism.

V 5 Bh Volume of a prism

V 5 (, · w)h The base is a rectangle, so B 5 , ? w.

V 5 (8 · 5)4 , 5 8, w 5 5, h 5 4

V 5 160 Simplify.

The volume is 160 cubic centimeters.

Find the volume of the cylinder. Use 3.14 for p. Round to thenearest tenth if necessary.

V 5 pr2h Volume of a cylinder

V 5 3.14 · 52 · 25 p < 3.14, r 5 5, h 5 25

V < 1,962.5 Simplify.

The volume is about 1,962.5 cubic feet.

Find the volume of each solid. Use 3.14 for p. Round to the nearesttenth if necessary.

1. 2. 3.

4. 5. 6.

5.8 m

4.1 m

7 m6.3 ft

9 ft11 mm

3 mm

3 mm

11 yd 5 yd

10 yd

4 m

12 m9 in.

2 in.

6 in.

The volume V of a cylinder with radius r is the area of the base B times the height h, or V 5 Bh. Sincethe base is a circle, the volume can also be written as V = pr2h, where B 5 pr2.

8 cm

5 cm

4 cm

The volume V of a prism or a cylinder is the area of the base B times the height h, or V 5 Bh.

5 ft

25 ft

Exercises

Example 1

Example 2


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Skills Practice


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1. 2. 3.

4. 5. 6.

7. rectangular prism: length, 6 in.; width, 4 in.; height, 13 in.

8. triangular prism: base of triangle, 9 cm; altitude 1 cm; height of prism,

15 cm

9. rectangular prism: length, 3.6 mm; width, 4 mm; height, 15.5 mm

10. triangular prism: base of triangle, 6 yd; altitude 5.9 yd; height of prism,

12 yd

11. cylinder: diameter, 8 m; height, 16.2 m

12. 13. 14.

4 yd

12 yd

6 yd

5 yd

12 m6 m

5 m

3 m

7 in.

6 in.

9 in.

18 in.

6 in.5 in.

4 in.

5 ft

4 ft

13 ft13

12 mi

10 mi12

14 yd

3.2 yd5 yd

4 cm

6.3 cm

10 cm

4 in.

13 in.7 m

5 m

9 m


Practice


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Find the volume of each solid. Use 3.14 for p. Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

7. rectangular prism: length, 10 m; width, 5 m; height, 5 m

8. triangular prism: base of triangle, 8 in; altitude, 8 in; height of prism, 6 in

9. cylinder: radius, 7 ft; height, 4 ft

10. cylinder: diameter, 6.4 cm; height, 4.9 cm

11. ALGEBRA Find the base of the triangle of a triangular prism with a height of 8 yards,

altitude of 4 yards, and a volume of 16 cubic yards.

12. ALGEBRA Find the height of a cylinder with a diameter of 5 meters, and a volume of

49.1 cubic meters.

13. WATER TANK About 7.5 gallons of water

occupy one cubic foot. About how many gallons

of water are in a cylindrical water tank with

dimensions shown in the figure?

4 m

5 m

1.1 yd

2.1 yd

0.8 yd

10 ft

4.2 ft

100 ft

40 ft

3 mm

12 mm

3 mm

7 in.

3 in.

2 in.7 cm

11 cm

4 cm


NAME ________________________________________ DATE ______________ PERIOD _____



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3. FOAM The figure below shows a piece

of foam packaging. Find the volume of

the foam.

1 ft

7 ft

3 ft

1 ft

2 ft2 ft

4. DONATIONS Lawrence is donating some

outgrown clothes to charity. The

dimensions of the box he is using are

shown below. How many cubic feet of

clothes will fit in the box?

2 ft

2.5 ft

3 ft

1. CAMPING A tent used for camping is

shown below. Find the volume of the

tent.

5 ft

6 ft

8 ft

2. CONSTRUCTION The dimensions of a

new tree house are shown below. How

many cubic feet of space will the tree

house contain?

2 m

6 m

3 m

5 m

23

5. FARM LIFE A trough used for watering

horses is shown in the figure. The

trough is half of a cylinder. How many

cubic feet of water will the trough hold?


15 ft

1 ft

6. FARM LIFE If the volume of the water in

the trough in Exercise 5 decreases by

5.6 ft3 per day, after how many days

will the trough be empty? Round to the

nearest tenth if necessary.


Enrichment

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Puzzling Patterns

In these visual puzzles, the challenge is to choose the one pattern that couldbe folded up into the box shown. You are not allowed to make any extra cutsin the patterns. The trick is that the six faces of the box must be arranged inthe correct order.

Circle the letter of the pattern that could be used to make each box.

1. A. B. C.

2. A. B. C.

3. A. B. C.

4. A. B. C.

5. A. B. C.


NAME ________________________________________ DATE ______________ PERIOD _____


Volume of Pyramids and Cones

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1. Compare the base areas and the heights of the two solids.

2. Fill the pyramid with rice, sliding a ruler across the top to level the

amount. Pour the rice into the cube. Repeat until the prism is filled. How

many times did you fill the pyramid in order to fill the cube?

3. What fraction of the cube’s volume does one pyramid fill?

Read the Lesson

4. How is the volume of a cone related to that of a cylinder?

5. How is the volume of a pyramid related to that of a prism?

6. Fill in the table about what you know from the diagram. Then compute

the volume of the pyramid.


7. Explain why the radius and height of cones and pyramids always form a

right angle.

6 in.8 in.

11 in.width of rectangle

length of rectangle

area of base

height of pyramid

volume of pyramid

NAME ________________________________________ DATE ______________ PERIOD _____



7-6

Example 1

Example 2

Volume Formulas

Pyramid Cone

V 5 }13

} Bh V 5 }13

} Bh

V 5 volume, h 5 height, V 5 volume, h 5 height, B 5 area of the base or ,w B 5 area of the base or pr 2


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Find the volume of the pyramid.

V 5 }13

} Bh Volume of a pyramid

V 5 }13

} s2h The base is a square, so B 5 s2.

V 5 }13

} · (3.6)2 · 9 s 5 3.6, h 5 9

V 5 38.88 Simplify.

The volume is 38.88 cubic meters.

Find the volume of the cone. Use 3.14 for the p.

V 5 }13

} pr2h Volume of a cone

V 5 }13

} · 3.14 · 52 · 10 p < 3.14, r 5 5, h 5 10

V < 261.7 Simplify.

The volume is about 261.7 cubic feet.


1. 2. 3.

4. 5. 6.5 in.

6 in.4 in.

7 m10 m

6 ft

4 ft4 ft

5 m7m

8 cm

5 cm

5 cm

4 yd

3 yd

5 ft

10 ft

9 m

3.6 m3.6 m

Exercises


Skills Practice


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1. 2. 3.

4. 5. 6.

7. 8. 9.

10. cone: diameter, 10 cm; height, 12 cm

11. triangular pyramid: triangle base, 20 mm; triangle height, 22 mm;

pyramid height, 14 mm

12. triangular pyramid: triangle base, 19 in.; triangle height, 21 in.; pyramid

height, 9 in.

13. cone: radius, 9.7 ft; height, 18 ft

8 mi12 mi

11 mi

13

5.4 in.

14 in.

15 in.9 m

3.5 m

A = 31 km2

11 km14 m

3 m

7 yd

5 yd8 yd

8 mm

10 mm7.4 km

4 km 14 km

6 ft

2 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Practice


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Find the volume of each pyramid. Use 3.14 for p. Round to the nearest tenth ifnecessary.

1. 2. 3.

Find the volume of each cone. Use 3.14 for p. Round to the nearest tenth ifnecessary.

4. 5. 6.

Find the volume of each solid. Use 3.14 for p. Round to the nearest tenth ifnecessary.

7. 8. 9.

10. PYRAMIDS The Great Pyramid has an astounding volume of about 84,375,000 cubic

feet above ground. At ground level the area of the base is about 562,500 square feet.

What is the approximate height of the Great Pyramid?

3 ft3 ft

5 ft

3 in.

2 in.

2 mm

8 mm8 mm

6 mm

3 ft

2 ft

4 ft

5 ft

2 yd1.5 yd

0.9 yd

18 mm

20 mm

10 in.

5 in.

1.6 cm

2.1 cm

1.2 cm3 yd

2 yd23

4 yd13




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3. AUTO REPAIR A funnel used to fill the

transmission on a car is shown below.

Find the volume of the funnel. Round


2 in.

9 in.

4. ART An artist created a

commemorative marker in the shape of

a triangular pyramid. Find the volume

of the stone used to make the marker.


12 ft

A = 15.6 ft2

1. DESSERT Find the volume of the ice

cream cone shown below. Round to the

nearest tenth if necessary.

1 in.

4 in.

2. SOUVENIRS On a trip to Egypt, Myra

bought a small glass pyramid as a

souvenir. Find the volume of the glass

used to make the pyramid. Round to

the nearest tenth.

4 in.

4 in.4 in.

5. FARMING The top of a silo is a cone, as

shown in the figure. Find the volume of

the cone. Round to the nearest tenth.

7 ft10 ft

6. LANDSCAPING When mulch was

dumped from a truck, it formed a

cone-shaped mound with a diameter of

15 feet and a height of 8 feet. What is

the volume of the mulch?


NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment7-6

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Two Truncated Solids

To create a truncated solid, you could start with an ordinary solid and thencut off the corners. Another way to make such a shape is to use the patternson this page.

The Truncated Octahedron

1. Two copies of the pattern at the right can be

used to make a truncated octahedron, a solid

with 6 square faces and 8 regular hexagonal

faces.

Each pattern makes half of the truncated

octahedron. Attach adjacent faces using glue or

tape to make a cup-shaped figure.

The Truncated Tetrahedron

2. The pattern below will make a truncated tetrahedron, a solid with

8 polygonal faces:

4 hexagons and 4 equilateral triangles.

Solve.

3. Find the surface area of the truncated

octahedron if each polygon in the pattern

has sides of 3 inches.

4. Find the surface area of the truncated

tetrahedron if each polygon in the pattern

has sides of 3 inches.

Tape orglue here.

Area Formulas for

Regular Polygons

(s is the length of one side)

triangle A 5 Ï3w

hexagon A 5 Ï3w

octagon A 5 2s2 (Ï2w 1 1)

3s2}2

s2}4


Spreadsheet Activity


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You can use a spreadsheet to calculate the volumes of pyramids and cones.

Use a spreadsheet to find the volume of the cone to the right.

Step 1 Recall that the formula for the volume of a cone is

V 5}1

3} pr2h.

Step 2 In cell A1, enter the radius of the cone and in cell B1,enter the height of the cone.

Step 3 In cell C1, enter an equals sign followed by 1/3*PI()*A1^2*B1. Then press ENTER to return the volume of the cone.

The volume of the cone is 56.54867 cubic inches.

Use a spreadsheet to find the volume of a pyramid with base 16square inches and height of 7 inches.

Step 1 Recall that the formula for the

volume of a pyramid is V 5 }1

3} Bh.

Step 2 In cell A2, enter the base of thepyramid and in cell B2, enter theheight of the pyramid.

Step 3 In cell C2, enter an equals signfollowed by 1/3*A2*B2. Then pressENTER to return the volume of the pyramid.

The volume of the pyramid is 37.33333 cubic inches.

Use a spreadsheet to find the volumes of each solid. Round to the nearest tenth if necessary.

1. pyramid: base, 3 m2; height, 10 m 2. pyramid: base, 15 cm2; height, 6 cm

3. cone: radius, 2 mm; height, 5 mm 4. cone: radius, 9 in.; height 8 in.

A1

3456

2

B C 3

76

16

Sheet 1 Sheet 2 Sheet 3

56.5486737.33333

3 in.

6 in.

Exercises

Example 1

Example 2


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1. Find the area of each face. Then find the sum of these areas.

2. Multiply the perimeter of a base by the height of the box. What does this

product represent?

3. Add the product from Exercise 2 to the sum of the areas of the two bases.

4. Compare your answers from Exercises 1 and 3.

Read the Lesson

5. Complete the sentence with the correct numbers. When you draw a net of

a triangular prism, there are ____ congruent triangular faces and ______

rectangular faces.

6. Explain how using a net helps to find the surface area of a figure.

7. If you unroll a cylinder, what does the net look like?


8. Surface area contains the word face. Remember to turn the object so that

each side faces you, and no face is left out. Fill in the chart to help you

remember how many faces you should be looking for in each figure.

Figure Total Number of Faces

rectangular prism

trianglular prism


Surface Area of Prisms and Cylinders


NAME ________________________________________ DATE ______________ PERIOD _____

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9 m

5 m

5 ft3 ft

7 ft

Exercises

Example 1

Example 2

Find the lateral and total surface areas of the rectangular prism.

Perimeter of Base Area of Base

P 5 2, 1 2w B 5 ,w

P 5 2(5) 1 2(3) B 5 5(3)

P 5 16 B 5 15

Use this information to find the lateral and total surface areas.

Lateral Surface Area Total Surface Area

L 5 Ph S 5 L 1 2B

L 5 16(7) or 112 S 5 112 1 2(15) or 142

The lateral surface area is 112 square feet and the total surface area of the prism is 142 square feet.

Find the surface area of the cylinder.Round to the nearest tenth.

Lateral Surface Area Total Surface Area

L 5 2prh S 5 L 1 2pr2

L 5 2 · 3.14(5)(9) S 5 282.6 1 2 · 3.14(5)2

L < 282.6 S < 439.6

The lateral area is about 282.6 square meters, and the surface area of thecylinder is about 439.6 square meters.

Find the lateral and total surface areas of each solid.Round to the nearest tenth if necessary.

1. 2. 3.

7 in.4 in.

8 in.

3 cm

5 cm

4 yd2 yd

5 yd

The lateral area L of a cylinder with height h and radius r is the circumference of the base times theheight, or , 5 2prh. The surface area S of a cylinder with height h and radius r is the lateral areaplus the area of the two bases, or S 5 , 1 2pr 2 or S 5 2prh 1 2pr 2.

The lateral area , of a prism is the perimeter P of the base times the height h of the prism, or , 5 Ph.The total surface area S of a prism is the lateral surface area , plus the area of the two bases 2B, orS 5 , 1 2B or S 5 Ph 1 2B.




NAME ________________________________________ DATE ______________ PERIOD _____

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Find the lateral and total surface areas of each solid. Use 3.14 for p.Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. cube: edge length, 11 m

14. rectangular prism: length, 9 cm; width, 13 cm; height, 18.4 cm

15. cylinder: radius, 9.4 mm; height, 15 mm

16. cylinder: diameter, 28 in.; height, 12.6 in.

18 in.

11 in.359 cm

15 cm

7 cm

12 cm

12

13 m

7 m9.4 m

10.4 ft

9 ft

11 km

5.7 km6 km

5 mi5 mi

12 mi

13 mi

4 m

8 m

10 m

7 cm

6.1 cm

6.1 cm

5 cm

17 cm

8 yd

7 yd

4 mm

3 mm

5 mm

7 mm

6 in.

3 in.6 ft

2 ft4 ft

Skills Practice



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Find the lateral and total surface areas of each solid. Use 3.14 for p.Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

7. ALGEBRA A rectangular prism has height 4 millimeters and width 5 millimeters. If the

total surface area is 166 square millimeters, what is the length of the prism?

8. WATER A cylindrical-shaped water storage tank with diameter 60 feet and height 20

feet needs to be painted on the outside. If the tank is on the ground, find the surface

area that needs painting.

9. CONCRETE Find the total surface area of the hollow concrete casing shown, including

the interior.

1 in.4 in.

5 in.

7 yd

5 yd

5 yd

8 yd

4 yd

8 in.

12 in.

8 in.

4 in.

9 cm

13 cm

3 m

2 m

2 m12

1.3 mm

0.8 mm

1.6 mm

1.1 mm

2.1 mm

5 ft

7 ft

Practice



NAME ________________________________________ DATE ______________ PERIOD _____

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3. FARMING Phil is planning to shingle

the roof on his barn shown below. How

many square feet will he be shingling?

41.6 ft

27 ft24 ft

24 ft

12 ft

4. FARMING Refer to Exercise 3. If one

package of shingles covers 325 square

feet, how many packages will Phil need

to buy?

1. BAKING The top and sides of the cake

shown below are to be covered in

frosting. Calculate the area that will be

covered with frosting.

12 in.9 in.

2 in.

2. GIFTS A birthday gift is placed inside

the box shown below. What is the

minimum amount of wrapping paper

needed to wrap this gift?

10 in.

14 in.7 in.

5. LIGHT SHOW A mirrored cylinder used

in a light show is shown below. Only

the curved side of the cylinder is

covered with mirrors. Find the area of

the cylinder covered in mirrors. Round


22 cm

30 cm

6. SOUP Emily has the flu, so she decides

to make chicken noodle soup. How

many square inches of metal were used

to make Emily’s can of soup? Round to

the nearest tenth.

3 in.

4 in.12




Enrichment

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Sliced Solids

In the diagrams on this page, a plane slices through a solid figure.The intersection of the plane with the solid is called a cross section.The drawings for each problem show a sliced solid and the dimensions of the resulting cross section.

Find the surface areas of the two solids that result from the slice.Use 3.14 for p. Round to the nearest tenth.

1. One-fourth of the cube is sliced off 2. One-third of the prism is sliced off

the top. the back.

3. The cube is sliced in half. 4. The cylinder is sliced in half.

5. The cylinder is sliced in half. 6. The prism is sliced in half.

100 ft

56.6 ft

40 ft40 ft

8 in.

8 in.

7 in.

9 in.

5 in.

10 cm

14.1 cm

16 m

20 m

80 m

100 m


NAME ________________________________________ DATE ______________ PERIOD _____


Surface Area of Pyramids

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1. Not including the base, how many faces does this pyramid have? What

shape are they?

2. How could you find the total area of the glass used for the building?

Read the Lesson

3. Complete the steps in finding the lateral and total surface areas of a pyramid.

Lateral Surface Area

L 5 }12

} P,

L 5 }12

}(36)(16)

L 5 288

Total Surface Area

S 5 L 1 B

S 5 1 92

The lateral surface area is square inches, and the total surface area

of the pyramid is 288 1 81 5 square inches.

4. What two areas are needed to calculate the surface area of a pyramid?

5. In a pyramid, what is the altitude of each face called?


6. Explain how the slant height of a pyramid is different from the height of

the pyramid. Find a real-life example of a solid pyramid and use it to

determine whether it is easier to measure the height or the slant height

of a pyramid. Explain your reasoning.

9 in.9 in.

16 in.

Find the lateral and total surface areas of the square pyramid.

Lateral Surface Area

L 5 }12

} P,

L 5 }12

}(16)(5) P 5 16, , 5 5

L 5 40

Total Surface Area

S 5 L 1 B

S 5 40 1 42

S 5 56

The lateral surface area is 40 square feet, and the total surface area of the pyramid is 56square feet.

Find the surface area of each solid. Round to the nearest tenth if necessary.

1. 2. 4 cm

5 cm5 cm

3 in.

2 in.2 in.

The lateral surface area L of a regular pyramid is half the perimeter P of the base times the slant

height ,, or L 5 }12

}P,. The total surface areas of a regular pyramid is the lateral area L plus the

area of the base B, or S 5 L 1 B or S 5 }12

}P, 1 B.

5 ft

4 ft4 ft


NAME ________________________________________ DATE ______________ PERIOD _____



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Exercises

Example


Skills Practice


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1. 2.

3. 4.

5. 6.

7. 8.

9. square pyramid: base side length, 4 cm; slant height, 7.3 cm

10. square pyramid: base side length, 5 yd; slant height, 12.7 yd

3 m

4 m4 m

A = 15.6 cm2

8.9 cm

6 cm6 cm

6 cm

10 cm

6 cm6 cm

11.5 yd

7 yd7 yd

A = 27.7 ft2

8 ft

8 ft8 ft

9 m

8 m8 m

A = 10.8 yd2

7 yd

5 yd

5 yd

5 yd

4 ft

6 ft6 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Practice


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Find the lateral and total surface areas of each regular pyramid. Round tothe nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. ALGEBRA A square pyramid has a lateral surface area of 20 square yards.

If the slant height is 2 yards, what is the total surface area of the

pyramid?

8. PYRAMIDS When the Great Pyramid was built, the slant height was about

610 feet and the length of the base was about 750 feet. Find the

approximate lateral surface area of the Great Pyramid when it was built.

2.1 cm2.1 cm

4.2 cm 8 ft

6.9 ft

8 ft8 ft

9 ft

3 yd

2.6 yd

3 yd

3 yd

3 yd

20 mm

16 mm

16 mm 32 in.

32 in.

2 m

1 m14

1 m14




NAME ________________________________________ DATE ______________ PERIOD _____

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3. HOBBIES When the butterfly net shown

below is fully extended, it forms the

shape of a pyramid with a slant height

of 26 inches. The sides of the square

base are 12 inches. Calculate the

amount of mesh material needed to

make the butterfly net.

4. HORTICULTURE The local college has a

greenhouse that is shaped like a square

pyramid, as shown below. The lateral

faces of the greenhouse are made of

glass. Find the surface area of the glass

on the greenhouse.

12 m

9 m9 m

1. ROOFS A farmer is planning to put new

roofing material on the pyramidal roof

of a work shed as shown below.

Calculate the number of square feet of

roofing material needed. Round to the

nearest tenth.

2. ROOFS Refer to Exercise 1. If the

roofing material costs $1.45 per square

foot, how much will it cost to put new

roofing material on the shed? Round to

the nearest cent.

5. ART Find the surface area of the

sculpture shown below.

6 m

4.4 m

4.4 m

6. COSTUMES The top of a costume hat is

shaped like a triangular pyramid, as

shown below. How much black felt is

needed to cover the sides of the

pyramid?

9 in.

11 in.

11 in. 11 in.

8 ft

10 ft10 ft

26 in.12 in.


NAME ________________________________________ DATE ______________ PERIOD _____

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Two Three-Dimensional Puzzles

In the nets on this page, segments of equal length are marked in the same way.

1. Make three copies of this pattern.

Use 2 inches for each side of the

central square.

Fold each pattern to make a pyramid.

Put the three pyramids together to

make a cube. Make a sketch of the

result.

2. Make four copies of this pattern.

Use 6 inches for the base of the figure.

Fold each pattern to make a solid. Put

the four solids together to make a regular

tetrahedron. Make a sketch of the result.

Solve.

3. Find the surface area of the cube in Exercise 1.

4. Find the volume of each of the three pyramids in Exercise 1.

5. Find the surface area of the tetrahedron in Exercise 2.

You will need to measure an altitude for one of the faces.


NAME ________________________________________ DATE ______________ PERIOD _____


Similar Solids

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1. If the model car is 4.2 inches long, 1.6 inches wide, and 1.3 inches tall,

what are the dimensions of the original car?

2. Make a conjecture about the radius of the wheel of the original car

compared to the model.

Read the Lesson

3. What is the scale factor for two similar solids?

4. If a 6-meter high pyramid is a model of an actual Egyptian pyramid and

the scale factor is }1

8}, what is the height of the actual pyramid?

5. A cube has a volume of 216 cubic feet. A smaller cube is similar by a scale

factor of 2. What is the length of a side of the smaller cube?


6. You can calculate the slant height of either pyramid on page 399 using

the Pythagorean Theorem by creating a right triangle with the pyramid’s

height, its slant height, and }1

2} of the side of the square base. The slant

height is the hypotenuse. The slant height of the larger pyramid is about

8.4 meters. How can you find the slant height of the smaller pyramid

without using the Pythagorean Theorem?


Similar solids have the same shape, their corresponding linear measures are proportional, and their corresponding faces are similar polygons.

The cones at the right are similar. Find the height of cone A.

}8

x} 5 }

4

3} Write a ratio.

4x 5 24 Find the cross products.

x 5 6 Simplify.

The height of the smaller cone is 6 inches.

The pyramids at the right are similar.Find the total surface area of pyramid B.

The scale factor }ab

} is }64

} or }32

}.

5 1}ab

}22

Write a proportion.

}98

S.4} 5 1}

32

}22

Substitute the known values. Let S represent the surface area.

}98

S.4} 5 }

9

4} 1}

32

}22

5 }32

} ? }32

} or }94

}

98.4 ? 4 5 9S Find the cross products.

}39

93.6} 5 }

99S} Divide each side by 9.

43.7 < S Simplify.

The surface area of pyramid B is approximately 43.7 square centimeters.

For Exercises 1 and 2, the solids in each pair are similar. Find the surface area of solid B.

For Exercises 3 and 4, find the value of x.

surface area of pyramid A}}}}surface area of pyramid B

NAME ________________________________________ DATE ______________ PERIOD _____


Similar Solids

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Exercises

Example 1

1. 2.

3. 4.

cone Bcone A

4 in.

3 in.

8 in.

solid A solid B

scale factor 5 5

S 5 24 units2

12

66

3

1.5 1.5

solid A

S 5 180 units2

solid B

166

24

x

x

515

3

Example 2

6 cm 4 cm

Pyramid AS 5 98.4 cm2

Pyramid BS 5


For Exercises 1–4, each pair of solids is similar. Find the volume ofsolid B.

1. 2.

3. 4.

For Exercises 5–12, find the measure of x. All pairs of figures are similar.

5. square pyramid A: base side 5 6 in., slant height 5 21 in.

square pyramid B: base side 5 x in., slant height 5 7 in.

6. cone A: base radius 5 8 cm, slant height 5 20 cm

cone B: base radius 5 x cm, slant height 5 15 cm

7. prism A: length 5 14 ft, width 5 12 ft, height 5 6 ft

prism B: length 5 3.5 ft, width 5 3 ft, height 5 x ft

8. regular triangle pyramid A: base side 5 3 in., slant height 5 10 in.

regular triangle pyramid B: base side 5 x in., slant height 5 25 in.

9. cylinder A: base radius 5 13 cm, length 5 8 cm

cylinder B: base radius 5 x cm, length 5 24 cm

10. prism A: length 5 7 ft, width 5 15 ft, height 5 8 ft

prism B: length 5 21 ft, width 5 x ft, height 5 24 ft

11. square pyramid A: base side 5 5 in., slant height 5 18 in.

square pyramid B: base side 5 x in., slant height 5 9 in.

12. cone A: base radius 5 16 m, height 5 28 m

cone B: base radius 5 x m, height 5 21 m

Skills Practice

Similar Solids

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V 5 8 units3

solid A solid B

scale factor 5 1.5

V 5 320 units3

solid A solid B

scale factor 51

2

V 5 4p cubic units3

solid A solid B

scale factor 5 2

V 5 324p units3

solid A solid B

scale factor 52

3


NAME ________________________________________ DATE ______________ PERIOD _____

Practice

Similar Solids

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Find the missing measure for each pair of similar solids. Round tothe nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. MODEL TRAINS The caboose of an N scale model train is 4}1

2} inches long. In the N scale,

1 inch represents 13}1

2} feet. What is the length of the original caboose?

8. ALGEBRA The volumes of two similar cylinders are 7 cubic meters and 56 cubic meters.

Find their scale factor.

For Exercises 9-11, use the similar prisms shown.

9. Write the ratio of the surface areas and the ratio

of the volumes of Prism B to Prism A.

10. Find the surface area of prism B.

11. Find the volume of prism A.

15 ft

9 ft3 ft

?

5.8 mm

8.7 mm

2.9 mm

2S = ? S = 288 mm

2 in.

2

3 in.

1 in.

S = ?S = 10 in

V = 9 m

5 m3 m

3 V = ?

8 yd4 yd

3V = 88 yd V = ?

S = 144 cm

V = 14 cm

4 cm2 cm

2

3

Prism B

4 cm

6 cm

1 cm

?


Similar Solids

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For Exercises 1–6, find the missing measure for each pair of similarsolids. Round to the nearest tenth if necessary.

3. BUILDING A room has dimensions that

are 12 ft 3 14 ft 3 9 ft. A larger room

is 1.5 times as large in each dimension.

What is the scale factor of the rooms'

volumes? (Hint: the scale factor of the

three-dimensional volumes is not the

same as the scale factor in one

dimension)

4. ART Ray takes a photo of a sculpture

he has just finished. In the photograph,

the sculpture is 4 inches wide. If each

inch in the photograph represents 2.5

feet, how wide is the sculpture?

1. ARCHITECTURE A model of a cylindrical

grain silo is 14 inches tall. On the

model 2 inches represents 5 feet. What

is the height of the actual grain silo?

2. AQUARIUMS A pet store has three sizes

of aquariums. The dimensions of the

smallest aquarium are 12 in. 3 16 in.

3 10 in. If other sizes of aquariums are

2 times and 2.5 times as large, what

are the dimensions of the other

aquariums?

5. MODELS An architectural model of a

skyscraper is shaped like a very tall

pyramid. The length of the sides of the

square base on the model are 6 inches

and the slant height is 24 inches. If the

scale factor of the model is }4

1

00}, what is

the slant height of the actual building?

6. CARS Sam has a picture of his favorite

type of car. In the photo, the car is 12

inches wide by 6 inches tall. If the

actual height of the car is 54 inches

tall, what is the actual length of the

car?

14 ft

9 ft

12 ft



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Ratios of Surface Area Parents often keep their babies bundled up with hats, blankets and extra layers of clothingto avoid a dangerous drop in body temperature due to heat loss. Babies are small and theyare more likely to suffer problems in very cold temperatures than adults. To understand thisbetter, medical researchers study the relationship between body surface area and bodyweight. For simplicity, we will approximate the surface area of infants and adults using asphere for the head and cylinders for the legs, arms, and torso.

1. Consider an infant who weighs 18 pounds and an adult who weighs 170 pounds.

Suppose the arms, legs, torso, and head can be approximated with the given solids

and dimensions below. Find the total surface area of both the infant and adult models.

Use 3.14 as an approximation for π.

2. What is the ratio of the total surface area to body weight of both the infant and the

adult? How are the two ratios related?

3. Based on your findings, why do you think it is important that parents bundle up their

babies when they are out in the cold? Write two or three sentences to explain your

reasoning.

r 5 4 in.

h 5 26 in.

r 5 6 in.

h 5 32 in.

r 5 2.5 in.

h 5 20 in.

r 5 2 in.

r 5 3 in.

r 5 3 in.

h 5 12 in.

r 5 1 in.

h 5 5 in.

r 5 1.5 in.

h 5 6 in.

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NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____Chapter 7 Quiz 1(Lessons 7-1, 7-2, and 7-3)

NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____Chapter 7 Quiz 2(Lessons 7-4 and 7-5)

7

Find the circumference and area of each circle.Use 3.14 for p. Round to the nearest tenth.

1. The diameter is 1}13

} yards. 1.

2. The radius is 4.8 centimeters. 2.

3. A total of 340 students were surveyed. If 15% of the students voted to get new school T-shirts, find the number of students who voted for the new shirts. 3.

4. Linda is a real estate agent and makes 6% of the sale price ofa house when it sells. If she sold a house for $140,000, how much did she make? 4.

5. Find the area of the figure. 5.6 cm

4 cm

4 cm

2 cm

Identify each solid. Name the number and shapes of the faces. Then name the number of edges and vertices.

1. 2. 1.

2.

3. Draw and label the top, front, and sideviews of the box shown.

3.Find the volume of each solid. Use 3.14 for p. Round to the nearest tenth ifnecessary.

4. 5. 4.

5.17 m

6 m

5.8 m

5 m

20 cm

11 cm


NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____Chapter 7 Quiz 3(Lessons 7-6 and 7-7)

7

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Chapter 7 Quiz 4(Lessons 7-8 and 7-9)

NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____7


1. 2. 1.

2.

Find the surface area of each solid. Use 3.14 for p. Round to the nearest tenth if necessary.

3. 4. 3.

4.

5. 5.

37 mm

14 mm

14 mm

12.1 mm

5 cm

9 cm

31.2 m

10.0 m15.1 m

6 cm

4 cm4 cm

10 ft

5 ft


1. 2. 1.

2.

3. Two similar cubes are shown.Find the missing side length. 3.

4. Two similar cubes are shown.Find the volume of the larger cube. 4.

5. Two cylinders are similar. The dimensions of the second cylinder are double those of the first. The volume of the first cylinder is 25 m3. Find the volume of the second cylinder. 5.

8 ft

6.2 ft6.2 ft

A = 6.9 in2

4 in.

4 in.4 in.

7.5 in.

3

33

V 5 27 in3

12

12?

V 5 ?

Chapter 7 Mid-Chapter Test(Lessons 7-1 through 7-5)


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SCORE _____7C

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

1. Find the circumference of a circle that has a radius of 10 inches.Round to the nearest tenth.

A. 62.8 in. B. 31.4 in. C. 6.28 in. D. 3.14 in. 1.

2. Find the area of a circle that has a radius of 3 centimeters. Round to the nearest tenth.

F. 9.7 in2 G. 18.8 in2 H. 28.3 in2 J. 113 in2 2.

3. Find the circumference of a circle that has a radius of 12 inches. Round to the nearest tenth. 3.

Find the area of each figure. Use 3.14 for p. Round to the nearest tenth if necessary.

4. 5. 4.

5.

6. Identify the solid to the right. Name the number and shapes of the faces. Then name the

number of edges and vertices. 6.

Find the volume of each solid. Round to the nearest tenth if necessary.

7.

A. 350 mm3 C. 1,050 mm3

B. 575 mm3 D. 823.5 mm3 7.

8.

F. 1,053 m3 H. 675.2 m3

G. 526.5 m3 J. 691.6 m3 8.

9. Find the volume of a cylinder that has a diameter of 6 inches and a height of 12 inches. Use 3.14 for p. Round to the nearest tenth. 9.

9 m

9 m

13 m

25 mm

6 mm

7 mm

31 mm

25 mm 4 cm


7

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Choose from the terms above to complete each sentence.

1. A(n) __________ is a set of points in a plane that are the same 1.distance from a given point in the plane, called the _________.

2. A two-dimensional flat surface that extends in all directions 2.is called a(n) __________.

3. A ___________ is a solid with flat surfaces that are polygons. 3.

4. Objects that are made up of more than one type of solid are called _________________. 4.

5. The distance from the center of a circle to any point on the 5.circle is called the __________, while the distance around the circle is called the __________.

6. The ratio of the circumference of a circle to its __________ 6.is always pi.

7. A(n) __________ is a point on a polyhedron where three or 7.more planes intersect.

8. A rectangular pyramid is a polyhedron with one base that is 8.a rectangle and four __________ that are triangles.

9. __________ is the measure of the space occupied by a solid. 9.

10. A __________ is a three-dimensional figure with one circular 10.base.

Define the term in your own words.

11. lateral face 11.

basecentercircle circumferencechordcomposite figurecomposite solidcone cylinder

diameteredge face lateral face lateral surface areapi (p) plane polyhedron prism

pyramid radius regular pyramidsimilar solidsslant heightsolid total surface area vertexvolume

Chapter 7 Vocabulary Test

NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____


Chapter 7 Test, Form 1

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1. Find the circumference of a circle that has a diameter of 6 inches.Round to the nearest tenth.

A. 18.8 in. B. 6 in. C. 9.4 in. D. 28.3 in. 1.

2. Find the area of a circle that has a radius of 4 inches. Round to the nearest tenth.

F. 3.2 in2 G. 4 in2 H. 50.2 in2 J. 12.6 in2 2.

For Questions 3, 4, and 5, find the area of each figure. Round to the nearest tenth if necessary.

3.

A. 615.8 in2

B. 44.0 in2

C. 22.0 in2

D. 153.9 in2 3.

4. 5.

F. 130 cm2 A. 64 mm2 4.

G. 114 cm2 B. 164.5 mm2

H. 94 cm2 C. 106.7 mm2

J. 122 cm2 D. 89.1 mm2 5.

6. Find the circumference of a circle with a diameter of 10 centimeters.Round to the nearest tenth.

F. 314.2 cm G. 78.5 cm H. 31.4 cm J. 15.7 cm 6.

7. Janie wants to leave a 15% tip. Her bill came to $24. How much tip should she leave?

A. $5.00 B. $3.00 C. $2.40 D. $3.60 7.

Use the solid shown on the right.

8. Identify the solid.

F. rectangular pyramid H. rectangular prism

G. pentagonal prism J. pentagonal pyramid 8.

9. Name the number of edges.

A. 6 B. 8 C. 12 D. 15 9.


10. 11.

F. 24 in3 A. 785.0 mm3 10.

G. 36 in3 B. 250 mm3

H. 12 in3 C. 261.8 mm3

J. 48 in3 D. 314.2 mm3 11.

5 mm

10 mm

4 in.

3 in.

6 in.

8 mm

8 mm

10 cm

13 cm6 cm

6 cm

4 cm

3 cm

7 in.


NAME ________________________________________ DATE ______________ PERIOD _____

Chapter 7 Test, Form 1 (continued)7

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12. 13.

F. 169.6 cm3 A. 480 in3 12.

G. 56.5 cm3 B. 117.3 in3

H. 37.7 cm3 C. 320 in3

J. 113.1 cm3 D. 960 in3 13.

For Questions 14–17, find the surface area of each solid. Round to the nearest tenth if necessary.

14. 15.

F. 100 in2 A. 172 m2 14.

G. 125 in2 B. 120 m2

H. 150 in2 C. 184 m2

J. 200 in2 D. 368 m2 15.

16. 17.

F. 96.6 in2 A. 864 ft2 16.

G. 108.0 in2 B. 504 ft2

H. 842.4 in2 C. 720 ft2

J. 1,123.2 in2 D. 414 ft2 17.

18. Two cylinders are similar. The dimensions of the first cylinder are halved.The volume of the first cylinder is 120 in3. Find the volume of the second cylinder.

F. 30 in3 G. 15 in3 H. 60 in3 J. 120 in3 18.

19. Two similar cylinders are shown.Find the missing radius length.

A. 1.5 m

B. 6 cm

C. 7 cm

D. 12 cm 19.

20. Two similar cylinders are shown.Find the missing volume.

F. 904.8 cm3

G. 452.4 cm3

H. 226.2 cm3

J. 6 cm3 20.

Bonus Find the area of the shaded B:region to the nearest tenth.

7 in.

5 in.

15 ft

12 ft12 ft

6 in.

6 in.

5.2 in.

6 in.

9 in.

10 m

6 m

5 m

5 m

4 m5 in.

5 in.

5 in.

12 in.

10 in.

8 in.3 cm

6 cm

?3 cm

8 cm4 cm

V 5 113.1 cm3 V 5 ?


Chapter 7 Test, Form 2A

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NAME ________________________________________ DATE ______________ PERIOD _____

SCORE _____7C

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1. Find the circumference of a circle that has a diameter of 8 inches.Round to the nearest tenth.

A. 8 in. B. 25.1 in. C. 12.6 in. D. 50.3 in. 1.

2. Find the area of a circle that has a radius of 6 inches. Round to the nearest tenth.

F. 6 in2 G. 28.3 in2 H. 18.8 in2 J. 113 in2 2.

For Questions 3, 4, and 5, find the area of each figure. Round to the nearest tenth if necessary.

3.

A. 1,134.1 ft2

B. 29.8 ft2

C. 59.7 ft2

D. 283.4 ft2 3.

4. 5.

F. 108 cm2 A. 111.7 in2 4.

G. 192 cm2 B. 168.3 in2

H. 150 cm2 C. 142.9 in2

J. 139.5 cm2 D. 96.1 in2 5.

6. Find the circumference of a circle with a radius of 7 centimeters. Round to the nearest tenth.

F. 22.0 cm G. 44.0 cm H. 615.8 cm J. 153.9 cm 6.

7. Susie needs to add 5% tax to her order. Her bill came to $120. How much tax will she pay?

A. $6.00 B. $12.00 C. $1.20 D. $3.60 7.

Use the solid shown below.

8. Identify the solid.

F. triangular prism H. rectangular pyramid

G. triangular pyramid J. rectangular prism 8.

9. Name the number of faces.

A. 5 B. 4 C. 3 D. 2 9.


10. 11.

F. 24.5 yd3 A. 82.5 yd3 10.

G. 180 yd3 B. 329.9 yd3

H. 270 yd3 C. 70.7 yd3

J. 540 yd3 D. 105.2 yd3 11.

5 yd

4.2 yd

9 yd

7.5 yd

8 yd

18.4 in.

3 in.

3 in.

8 in.

12 in.

9 cm

7 cm

12 cm

19 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Chapter 7 Test, Form 2A (continued)7

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12. 13.

F. 813.7 yd3 A. 117.3 in3 12.

G. 2,309.1 yd3 B. 126 in3

H. 329.9 yd3 C. 42 in3

J. 769.3 yd3 D. 52 in3 13.

14. PAINTING A water storage tank shown at the right is to be painted. What is the surface area to be painted? Assume that the bottom does not need painting.

F. 4,082.0 ft2 H. 18,849.6 ft2

G. 4,398.2 ft2 J. 6,283.2 ft2 14.

For Questions 15–17, find the surface area of each solid. Round to the nearest tenth if necessary.

15. 16.

A. 301.2 cm2 F. 312.3 ft2 15.

B. 78.0 cm2 G. 467.6 ft2

C. 226.6 cm2 H. 631.0 ft2

D. 234.0 cm2 J. 1,588.6 ft2 16.

17.

A. 126 cm2 B. 276 cm2

C. 120 cm2 D. 156 cm2 17.

18. Two cylinders are similar. The dimensions of the first cylinder are triple those of the second. The volume of the first cylinder is 2,700 in3. Find the volume of the second cylinder.

F. 300 in3 G. 27 in3 H. 100 in3 J. 120 in3 18.

19. Two similar cylinders are shown. Find the missing height.

A. 3 cm C. 27 cm

B. 9 cm D. 13 cm 19.

20. Two similar cylinders are shown.Find the missing volume.

F. 3053.7 cm3 H. 200 cm3

G. 432 cm3 J. 9 cm3 20.

Bonus The surface area of a cube is 1,350 cm2. Find the B:length of an edge.

10 cm

6 cm6 cm

14.6 ft

10.2 ft

6 cm

6 cm

6 cm

5.2 cm

15 cm

20 ft

60 ft

7 in.

4 in.9 in.

7 yd

15 yd

?

2 cm

6 cm

9 cm

V 5 113.1 cm3 V 5 ?

WHJH Math

math 8 - woodland hills school district · skills practice: level 1 practice: ... to find surface...

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