lahge 10 csse cp 500-507.qxd 4/7/09 5:51 am page 500 9.4 volume of prisms and...

The amount of water in an aquarium is an example of volume. Theof a solid is the number of cubic units contained in its interior. volume

Words Volume � (area of base)(height)

Symbols V � Bh

VOLUME OF A PRISM

Volume of a Prism The process used in Example 1 can be used todetermine the volume of any prism.

500 Chapter 9 Surface Area and Volume

Goal

Find the volumes of prismsand cylinders.

Key Words

• prism p. 483

• cylinder p. 485

• volume

9.49.4 Volume of Prisms and Cylinders

Find the volume of the box by determining how many unit cubes fit in the box.

Solution

The base is 5 units by 3 units. So, 3 p 5, or 15 unit cubes are needed to cover the base layer.

There are 4 layers. Each layer has 15 cubes. So, the total number of cubes is 4 p 15, or 60.

ANSWER � The volume of the box is 60 cubic units.

EXAMPLE 1 Find the Volume of a Rectangular Prism

5 units3 units

4 units

unit cube

1 11

h

B

B

h

� �

Volume of prism � area of base � height

READING TIP

Volume is measured incubic units, such as ft3,read as “cubic feet.”

Student Help

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Find the volume of the prism.

1. 2. 3. 7 in.

10 in.

7 in.5 cm

5 cm5 cm

6 ft

4 ft9 ft

9.4 Volume of Prisms and Cylinders 501

STUDY TIP

Because you aremultiplying three unitsof measure when youfind volume, youranswer will always be in cubic units.ft � ft � ft � ft3

Student Help

Find Volume of Prisms

Find the volume of the prism.

a. b.

Solution

a. V � Bh Write the formula for volume of a prism.

� (7 p 4) p 5 Area of rectangular base � l p w � 7 p 4.

� 140 Simplify.

ANSWER � The volume is 140 cubic inches.

b. V � Bh Write the formula for volume of a prism.

� ��12

� p 8 p 6� p 3 Area of triangular base � �12� p 8 p 6.

� 72 Simplify.

ANSWER � The volume is 72 cubic feet.

3 ft

6 ft8 ft

5 in.

4 in.7 in.

EXAMPLE 222 Find the Volume of a Prism

Volume of a Cylinder The method for finding the volume of a cylinder is the same for finding the volume of a prism.

� �

r

B � πr 2

h

Volume of cylinder � area of base � height

cgpe-0904 8/13/01 12:55 PM Page 501


Words Volume � (area of base)(height)

Symbols V � Bh

� πr 2h

VOLUME OF A CYLINDER

a. How do the radius and height of the mug compare to the radius and height of the dog bowl?

b. How many times greater is the volume of the bowl than the volume of the mug?

Solution

a. The radius of the mug is 2 inches and the radius of the dog bowl is 6 inches. The radius of the bowl is three times the radius of the mug.The height of the mug is the same as the height of the bowl.

b. Volume of mug

V � πr 2h Write the formula for volume.

� π(22)(4) Substitute for r and for h.

� 16π Simplify.

To compare the volume of the bowl to the volume of the mug, dividethe volume of the bowl by the volume of the mug.

�VV

oo

lluu

mm

ee

oo

ff

bm

ouwgl

�� 11464ππ

� � 9

ANSWER � The volume of the bowl is nine times the volume of themug.

EXAMPLE 333 Compare Volumes of Cylinders

h

B r

Find the volume of the cylinder. Round your answer to the nearest

whole number.

4. 5. 6. 4 m

10 m

1 in.

5 in.3 ft

2 ft

Find Volume of Cylinders

6 in.

4 in.

2 in.

4 in.

Volume of dog bowl

V � πr 2h

� π(62)(4)

� 144π

cgpe-0904 8/13/01 12:55 PM Page 502

Based upon the units, tell whether the number is a measure of

surface area or volume.

1. 5 ft3 2. 7 yd2 3. 3 m2 4. 2 cm3

Candles Find the volume of the candle.

5. 6. 7.

Using Unit Cubes Find the number of unit cubes that will fit in the

box. Explain your reasoning.

8. 9. 10.

Volume of a Prism Find the volume of the prism.

11. 12. 13.

Volume of a Cube In Exercises 14–16, you are given the length

of each side of a cube. Sketch the cube and find its volume.

14. 3 meters 15. 7 feet 16. 10 centimeters

In Exercises 17 and 18, make a sketch of the solid.

Then find its volume.

17. A prism has a square base with 4 meter sides and a height of 7 meters.

18. A prism has a rectangular base that is 3 feet by 6 feet and a height of 8 feet.

Visualize It!

12 m

9 m4 m

3 cm

6 cm

2 cm5 in.

4 in.

5 in.

4

23

3

44

2

35

Practice and Applications

12 cm

B ≈ 23.4 cm2

6 cm8 cm

10 cm

B ≈ 63.6 cm2

12 cm

Skill Check

Vocabulary Check

Guided Practice


Exercises9.49.4

Example 1: Exs. 8–10Example 2: Exs. 11–18Example 3: Exs. 27–40

Homework Help

Extra Practice

See p. 692.

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Finding Volume Find the volume of the combined prisms.

19. 20. 21.

Shopping In Exercises 22–24, use the information about the sizes of

the cereal boxes shown below.

22. Find the volume of each box of cereal.

23. How many small boxes of cereal do you have to buy to equal the amount of cereal in a large box?

24. Which box gives you the most cereal for your money? Explain.

Soo Locks In Exercises 25 and 26, use the information below.

Lake Superior is about 22 feet higher than Lake Huron. In order forships to safely pass from one lake to the other, they must go throughone of the four Soo Locks.

25. Water is added to the MacArthur Lock until the height is increased by22 feet. To find the amount of water added to the lock, find the volumeof a rectangular prism with a length of 800 feet, a width of 80 feet, and aheight of 22 feet.

26. How many gallons of water are added to the MacArthur Lock to raise the ship to the level of Lake Superior? Use the fact that 1 ft3 ≈ 7.5 gal.

2 m

5 m

7 m

10 m

4 m

2 in.

6 in.

1 in.

1 in.

4 in.

2 in.

2 ft 5 ft

3 ft

2 ft8 ft


LakeSuperiorLake

Huron

22 ft800 ft

Not drawn to scale

LakeSuperior

LakeHuron

Top View

Side Viewlower gates upper gates

80 ft

Nutrition Fa

cts

Serving Size 1

bowl

Serving per

container

Amount per s

erving

Calories 45

%

Daily Value

Total Fat 0

g

Saturated F

at 0g

Cholesterol

0mg

Sodium 0m

g

Carbohydr

ate 2g

Dietary Fib

er 0g

Sugars 0g

Protein les

s than 1g

Vitamin A ***

Calcium ***

Vitamin C ***

Iron ***

*****

********

********

**

********

********

**

8 in. 2 in.

10 in.

$2.00Whole Whole

Cereal

Part of a well balanced breakfast

10 in.

16 in.

4 in.

$6.00 Serving per c

ontainer

Calories 45

%Daily V

alue

Cholesterol

0mg

Dietary Fiber

0g

Sugars 0

Calcium ***

Vitamin C ***

Iron ***

*****

**********

********

**********

********

Whole Whole

Cereal

Part of a well balanced breakfast

HOMEWORK HELP

Extra help with problemsolving in Exs. 19–21 is at classzone.com

Student HelpI C L A S S Z O N E . C O M

SOO LOCKS The first lockssystem between Lake Superiorand Lake Huron was builtaround 1797. Today, four locksare available in the Soo Lockssystem.

Civil Engineering

cgpe-0904 8/13/01 12:55 PM 04

Volume of a Cylinder Find the volume of the cylinder. Round your

answer to the nearest whole number.

27. 28. 29.

Swimming Pools In Exercises 30–32, find the volume of the pool.

Round your answer to the nearest whole number.Then compare the

volumes of the pools to answer Exercise 33.

30. 31. 32.

33. Which pool above requires the least amount of water to fill it?

In Exercises 34 and 35, use the information below.

Suppose that a 3-inch by 5-inch index card is rotated around a horizontal line and a vertical line to produce two different solids.

34. Find the volume of each solid.

35. Which solid has a greater volume? Explain your reasoning.

Aquariums In Exercises 36 and 37, use the information below.

The Giant Ocean Tank at the New England Aquarium is a cylinder that is 23 feet deep and 40 feet in diameter as shown.

36. Find the volume of the tank.

37. How many gallons of water are needed to fill the tank? (1 ft3 ≈ 7.5 gal)

38. Personal Aquariums To avoid overcrowding in a personalaquarium, you should buy one fish for every gallon of water (231 in.3 ≈ 1 gal). About how many fish should be in an aquarium that is a rectangular prism measuring 20 incheswide, 10 inches long, and is filled with water to a height of 11 inches?

Visualize It!

15 ft

3 ft

24 ft

4 ft4 ft

20 ft

12 m6 m

9 in.

4 in.


5 in.

3 in.3 in.

5 in.

40 ft

23 ft

AQUARIUM DIVER Inaddition to feeding and takingcare of the fish and the plantsin an aquarium, divers makesure that the tank does notget too crowded.

Careers

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Find the volume of the passenger car of the Space Spiral at Cedar Point Amusement Park in Sandusky, Ohio.

Solution

The passenger car is a cylinder with a “hole” in it. To find the volume, subtract the volume of the hole from the volume of the larger cylinder.

Volume of larger cylinder � πr 2h

� π(102)(14)

≈ 4398

Volume of “hole” � πr 2h

� π(42)(14)

≈ 704

ANSWER � The volume of the passenger car is about 4398 � 704 � 3694 cubic feet.

In Exercises 39 and 40, use the cartons shown.

39. Find the volume of each carton of ice cream.

40. Terry assumes that because thedimensions doubled, the jumbocarton contains twice as much icecream as the regular carton. Is Terry right? Explain your reasoning.

You be the Judge

Finding Volume In Exercises 41–43, find the volume of the solid.

41. 42. 43.

Using Algebra Write an expression for the volume of the solid in

terms of x.

44. 45. 46. x

5x3x

x

2x

7

4

x3

4 m

4 m

4 m

1 m3 ft

8 ft

10 ft8 in.

2 in.

2 in.

6 in.

1 in.


EXAMPLE Find Volume

10 cm

5 cm

CCooooll

20 cm JUMBO

10 cm

CCooooll

HOMEWORK HELP

Extra help with problemsolving in Exs. 41–43 is at classzone.com

Student HelpI C L A S S Z O N E . C O M

14 ft

10 ft 4 ft

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Challenge In Exercises 47–49, find the missing dimension(s). If

necessary, round your answer to the nearest whole number.

47. A cylinder has a volume of 100.48 cubic inches and a diameter of 4 inches. Find the height of the cylinder.

48. A cylinder has a volume of 1538.6 cubic feet and a height of 10 feet. Find the radius of the cylinder.

49. The length of a rectangular prism is twice its width. The height of the prism equals the width. Find the dimensions of the prism, given that the volume is 54 cubic inches.

50. Multiple Choice What is the approximate volume of the cylindershown at the right?

�A 100 in.3 �B 785 in.3

�C 1570 in.3 �D 6280 in.3

51. Multiple Choice The volume of the prism shown at the right is168 cubic feet. What is the height of the prism?

�F 6 feet �G 7 feet

�H 8 feet �J 9 feet

Using the Pythagorean Theorem Find the unknown side length.

Round your answer to the nearest tenth. (Lesson 4.4)

52. 53. 54.

Surface Area Find the surface area of the solid. If necessary, round

your answer to the nearest whole number. (Lessons 9.2, 9.3)

55. 56. 57.

Solving Equations Solve the equation. (Skills Review, p. 672)

58. x � 7 � 0 59. m � 1 � �12 60. 10 � c � �3

61. �34

�b � 24 62. �14d � 2 63. 6n � 102

Algebra Skills

5 yd

12 yd8 m

9 m2 m

7 ft

3 ft

a

18 14b

8

9a

7 12

Mixed Review

Standardized Test

Practice

5 in.

20 in.

x

3 ft7 ft

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Activity Investigating VolumeInvestigating VolumeInvestigating Volume9.59.5

Question

How does the volume of a pyramid relate to the volume of a

prism with the same base?

Explore

�1 Use a ruler to draw the two nets shown below on poster board. Be sure to draw the dashed lines on the net as shown.

�2 Cut out the nets. �3 Fold along the dashed linesto form an open prism andan open pyramid.

�4 Tape each solid to hold it inplace, making sure that theedges do not overlap.

• poster board• ruler• scissors• tape• unpopped popcorn,

uncooked rice, ordried beans

Materials

1 716 in.

2 in.

2 in.

2 in. 2 in.

cgpe-0905 8/13/01 1:07 PM Page 508

Activity 509

Explore

Think About It

1. Compare the area of the base of the pyramid to the area of thebase of the prism. Fitting the pyramid inside the prism will help.What do you notice?

2. Compare the heights of the solids. What do you notice?

3. Which solid do you think has the greater volume? Why?

Think About It

4. How many times did you empty the contents of the pyramid intothe prism? What does this tell you about the volume of thepyramid and the prism?

5. Complete each sentence with the appropriate number.

Volume of the prism � __?__ � Volume of the pyramid

Volume of the pyramid � __?__ � Volume of the prism

6. Use your results from Exercise 5 to write a formula for the volumeof a pyramid.

Volume of a prism: V � Bh

Volume of a pyramid: V � __?__ Bh

�5 Fill the pyramid to the top withunpopped popcorn, uncookedrice, or dried beans. Pour thecontents of the pyramid into theprism.

�6 Repeat Step 5 until the prism isfull. Count the number of timesyou empty the contents of thepyramid into the prism.

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lahge 10 csse cp 500-507.qxd 4/7/09 5:51 am page 500 9.4 volume of prisms and...

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