9.4 Volume of Prisms and Cylinders

Goal
Find the volumes of prisms
and cylinders.
Key Words
• prism p. 483
• cylinder p. 485
• volume
Student Help
READING TIP
Volume is measured in
cubic units, such as ft 3 ,
read as “cubic feet.”
9.4 Volume of Prisms
and Cylinders
500 Chapter 9 Surface Area and Volume
The amount of water in an aquarium is an example of volume. The
volume of a solid is the number of cubic units contained in its interior.
EXAMPLE 1 Find the Volume of a Rectangular Prism
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.
Volume of a Prism The process used in Example 1 can be used to
determine the volume of any prism.
VOLUME OF A PRISM
Words Volume (area of base)(height)
Symbols V Bh
5 units
unit cube
Volume of prism area of base height
B
B
1
4 units
3 units
h
1
1
h

Student Help
STUDY TIP
Because you are
multiplying three units
of measure when you
find volume, your
answer will always
be in cubic units.
ft ft ft ft 3
EXAMPLE 2 Find the Volume of a Prism
Find the volume of the prism.
a. b.
5 in.
7 in.
Solution
4 in.
a. V Bh Write the formula for volume of a prism.
(7 p 4) p 5 Area of rectangular base lpw 7 p 4.
140 Simplify.
ANSWER The volume is 140 cubic inches.
b. V Bh Write the formula for volume of a prism.
1 p 8 p 6 p 3
2
Area of triangular base 1 p 8 p 6.
2
72 Simplify.
ANSWER The volume is 72 cubic feet.
Find Volume of Prisms
Find the volume of the prism.
1.
6 ft
2.
5 cm
3.
7 in.
7 in.
9 ft
4 ft
5 cm
5 cm
10 in.
Volume of a Cylinder The method for finding the volume of
a cylinder is the same for finding the volume of a prism.
Volume of cylinder area of base height
r
B πr 2
8 ft
6 ft
3 ft
9.4 Volume of Prisms and Cylinders 501
h

502 Chapter 9 Surface Area and Volume
VOLUME OF A CYLINDER
Words Volume (area of base)(height)
Symbols V Bh
πr 2 h
EXAMPLE 3 Compare Volumes of Cylinders
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 2 h Write the formula for volume.
π(2 2 )(4) Substitute for r and for h.
16π Simplify.
To compare the volume of the bowl to the volume of the mug, divide
the volume of the bowl by the volume of the mug.
Volume
of
bowl
Volume
of
mug
144π
9
16π
ANSWER The volume of the bowl is nine times the volume of the
mug.
Find Volume of Cylinders
Find the volume of the cylinder. Round your answer to the nearest
whole number.
4. 2 ft 5. 1 in.
6. 4 m
3 ft
5 in.
4 in.
2 in.
4 in.
B
r
10 m
h
6 in.
Volume of dog bowl
V πr 2 h
π(6 2 )(4)
144π

9.4
Exercises
Guided Practice
Vocabulary Check
Based upon the units, tell whether the number is a measure of
surface area or volume.
1. 5 ft 3
Practice and Applications
Extra Practice
See p. 692.
Skill Check
Homework Help
Example 1: Exs. 8–10
Example 2: Exs. 11–18
Example 3: Exs. 27–40
2. 7 yd 2
Candles Find the volume of the candle.
3. 3 m 2
5. 6.
6 cm
7.
8 cm
12 cm
10 cm
B ≈ 63.6 cm 2
4. 2 cm 3
Using Unit Cubes Find the number of unit cubes that will fit in the
box. Explain your reasoning.
8. 9. 10.
2
3
3
5
4
4
Volume of a Prism Find the volume of the prism.
11. 12. 13.
5 in.
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
Visualize It!
5 in.
4 in.
2 cm
6 cm
3 cm
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.
12 m
18. A prism has a rectangular base that is 3 feet by 6 feet and a height
of 8 feet.
4 m
3
12 cm
B ≈ 23.4 cm 2
9 m
9.4 Volume of Prisms and Cylinders 503
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4

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HOMEWORK HELP
Extra help with problem
solving in Exs. 19–21 is
at classzone.com
Civil Engineering
SOO LOCKS The first locks
system between Lake Superior
and Lake Huron was built
around 1797. Today, four locks
are available in the Soo Locks
system.
504 Chapter 9 Surface Area and Volume
Finding Volume Find the volume of the combined prisms.
19.
3 ft
20. 6 in.
21. 2 m
2 in.
5 m
8 ft
1 in.
2 ft
4 in.
2 ft 5 ft
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 for
ships to safely pass from one lake to the other, they must go through
one of the four Soo Locks.
Lake
Huron
22 ft
Lake
Huron
2 in.
25. Water is added to the MacArthur Lock until the height is increased by
22 feet. To find the amount of water added to the lock, find the volume
of a rectangular prism with a length of 800 feet, a width of 80 feet, and a
height of 22 feet.
1 in.
8 in.
Top View
lower gates
80 ft
Side View
upper gates
800 ft
$2.00 $2.00
Not drawn to scale
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 ft 3 ≈ 7.5 gal.
Whole Whole
Grain Grain
Cereal
Part of a well balanced breakfast
Nutrition Facts
Serving Size 1 bowl
Serving per container
Amount per serving
Calories 45
%Daily Value
Total Fat 0g
Saturated Fat 0g
Cholesterol 0mg
Sodium 0mg
Carbohydrate 2g
Dietary Fiber 0g
Sugars 0g
Protein less than 1g
Vitamin A ***
Calcium ***
Vitamin C ***
Iron ***
*****
******************
******************
10 in.
2 in.
Whole Whole
Grain Grain
7 m
10 in.
$6.00
Cereal
Part of a well balanced breakfast
Nutrition Facts
Serving Size 1 bowl
Serving per container
Amount per serving
Calories 45
%Daily Value
Total Fat 0g
Saturated Fat 0g
Cholesterol 0mg
Sodium 0mg
Carbohydrate 2g
Dietary Fiber 0g
Sugars 0
Protein less than 1g
Vitamin A ***
Calcium ***
Vitamin C ***
Iron ***
*****
******************
******************
10 m
16 in.
4 in.
4 m
Lake
Superior
Lake
Superior

Careers
AQUARIUM DIVER In
addition to feeding and taking
care of the fish and the plants
in an aquarium, divers make
sure that the tank does not
get too crowded.
Volume of a Cylinder Find the volume of the cylinder. Round your
answer to the nearest whole number.
27.
4 in.
28.
6 m
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.
20 ft
31.
24 ft
32.
4 ft
4 ft
3 ft
33. Which pool above requires the least amount of water to fill it?
Visualize 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.
3 in.
9 in.
5 in.
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.
23 ft
36. Find the volume of the tank.
37. How many gallons of water
are needed to fill the tank?
(1 ft 3 ≈ 7.5 gal)
15 ft
38. Personal Aquariums To avoid overcrowding in a personal
aquarium, 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 inches
wide, 10 inches long, and is filled with water to a height of
11 inches?
3 in.
5 in.
40 ft
12 m
9.4 Volume of Prisms and Cylinders 505

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Extra help with problem
solving in Exs. 41–43 is
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You be the Judge
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 2 h
π(10 2 )(14)
≈ 4398
Volume of “hole” πr 2 h
π(4 2 )(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 the
dimensions doubled, the jumbo
carton contains twice as much ice
cream as the regular carton. Is
Terry right? Explain your reasoning.
Finding Volume In Exercises 41–43, find the volume of the solid.
41. 2 in. 1 in. 42. 8 ft 43. 1 m
3 ft
6 in.
506 Chapter 9 Surface Area and Volume
EXAMPLE Find Volume
Using Algebra Write an expression for the volume of the solid in
terms of x.
44. 45. 46. x
3
x
4
2 in.
8 in.
x
2x
10 cm
10 ft
7
Cool
5 cm
20 cm
3x
14 ft
4 m
JUMBO JUMBO
Cool
10 ft 4 ft
5x
4 m
10 cm
4 m

Standardized Test
Practice
Mixed Review
Algebra Skills
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 cylinder
shown at the right?
20 in.
A 100 in. 3
C 1570 in. 3
B 785 in. 3
D 6280 in. 3
51. Multiple Choice The volume of the prism shown at the right is
168 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.
8
54.
7
12
a
9
b
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.
3 ft
56. 57.
7 ft
2 m
Solving Equations Solve the equation. (Skills Review, p. 672)
58. x 7 0 59. m 1 12 60. 10 c 3
61. 3
b 24 62. 14d 2 63. 6n 102
4
9 m
8 m
7 ft
12 yd
5 yd
9.4 Volume of Prisms and Cylinders 507
18
a
x
3 ft
14
5 in.