160 CHAPTER 4
Geometry
Textbook Reference Section 8.4
4.4 VOLUME AND SURFACE AREA
CLAST OBJECTIVES
" Calculate volume and surface area
" Infer formulas for measuring geometric figures
" Select applicable formula for computing measures of geometric figure
" Solve real-world problems involving volume and surface area
Volume is space occupied inside a container. It is always in cubic units or units cubed.
Name
Regular Shape
Volume Formula
s
s
Cube
( side )
s
3
= s
3
L
W
Rectangular Solid
L × W × H
H
r
Cylinder
2
h
π× r × h
r
Cone
h
Sphere
1
2
×π×r ×h
3
4
3
×π×r
3
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SECTION 4.4
Examples
a) Find the volume.
4m
9m
b) Find the volume of a sphere of radius 8
feet.
Volume and Surface Area 161
Solutions
Volume (V) of a cylinder = π × r 2 × h
V = π × (4 m) 2 × (9 m)
V = π × 16 m 2 × 9 m
V = 144 π m 3
Volume (V) of a sphere =
4
× π × r3
3
4
× π × ( 8 ft) 3
3
4
V = × π × 512 ft 3
3
2048
π ft 3
V=
3
2
V = 682 π ft 3
3
V=
c) The volume of a cube is 125 cubic
inches. What is the measure of one side?
Volume (V) of a cube = s
3
3
V=s
3
125 in 3 = s
3
125 in
3
=
3
s
3
5 in = s
d) Find the volume of a cone of radius 3
millimeters and height 5 millimeters.
Volume (V) of a cone =
1
2
×π×r ×h
3
1
2
× π × (3 mm) × 5 mm
3
1
2
= × π × 9 mm × 5 mm
3
1
3
= × π × 45 mm
3
45
3
=
π mm
3
3
= 15 π mm
V=
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162 CHAPTER 4
Geometry
Check Your Progress 4.4
Find the volume.
1.
3 ½ ft
2.
3m
1.5 m
3 ½ ft
7.2 m
3 ½ ft
3.
2 cm
4.
3"
9 cm
10"
5. Find the volume of a cube of side 5.2 ft.
6. Find the volume of a rectangular solid of length 7
1
1
cm, width 2 cm, and height of
2
3
4 cm.
7. Find the volume of a cylinder of height 22 inches and diameter 8 inches.
8. Find the volume of a cone of height 9 cm and radius 2 cm.
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SECTION 4.4
Volume and Surface Area 163
9. Find the volume of a sphere of radius 3 ft.
10. A swimming pool is 12 ft wide, 20 ft long, and 8 ft deep. What is the volume of
water in the pool?
1
11. A soda can is 6 inches tall and has a radius of 1 inches. How much soda is in a full
2
can?
12. What is the volume of sand, in cm 3, that can be contained in a shoe box of
dimensions 10 cm by 6 cm by 18 cm?
13. A ball has radius of 6 inches. How much water, in cubic inches, can the ball hold?
14. A party hat is in the shape of a cone. Jane loves sugar. How much sugar, in cubic
inches, can Jane put in her hat if it is 9 inches tall with a radius of 2 inches?
Surface Area
Surface area is the total area of all surfaces on a container. It is always measured in square
units or units squared.
•
•
A cube has six surfaces. Each surface is a square of similar area.
A rectangular solid has six surfaces also.
o Two surfaces have area: A = L × W
o Two surfaces have area: A = W × H
o Two surfaces have area: A = L × H
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164 CHAPTER 4
Geometry
Name
Regular Shape
Surface Area Formula
s
6 × ( side ) 2 = 6 s 2
Cube
s
L
2 LW + 2 WH + 2 LH
Rectangular Solid
H
W
Examples
e) Find the surface area.
4.2 ft
4.2 ft
Solutions
Surface area (S) of a cube = 6 s 2
S = 6 × ( 4.2 ft) 2
S = 6 × 17.64 ft 2
S = 105.84 ft 2
4.2 ft
f) Find the surface area of a box with
dimensions 10 cm by 8 cm by 3 cm.
Note that the dimensions are different
values. This indicates that the box is
rectangular in shape.
S = 2 LW + 2 WH + 2 LH.
Let H = 10 cm, W = 8 cm, and L = 3 cm
S = 2(3 cm × 8 cm) + 2(8 cm × 10 cm)
+ 2(3 cm × 10 cm)
S = 2(24 cm 2 ) + 2(80 cm 2) + 2(30 cm 2)
S = 48 cm 2 + 160 cm 2 + 60 cm 2
S = 268 cm 2
Check Your Progress 4.4
Find the volume.
15.
8.7 cm
16.
3'
2'
8.7 cm
10'
8.7 cm
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SECTION 4.4
Volume and Surface Area 165
17. Mary wants to cover a box that measures 20 cm by 10 cm by 5 cm. How much paper,
in sq. cm., does Mary need?
18. A cylindrical container can be formed from two tops and a rectangle as shown.
Based on the figure, what is the formula for calculating the surface area ( S ) of the
cylinder?
r
r
r
h
h
See If You
Remember
SECTIONS 4.1 – 4.3
Find the perimeter.
2 in
2.
1.
6.3 m
7 in
1 in
6 in
1
2
3. Find the distance around a rectangle of length 4 cm and width 2 cm.
2
3
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166 CHAPTER 4
Geometry
4. A circular track has a diameter of 50 yards. What is the distance around the track?
Find the area of each figure.
5.
|
6.
5m
__
__
|
3m
|------------------- 7 m ------------------|
3.3 cm
7.
8 ft
3 ft
8. Find the area of a circle of diameter 30 inches.
9. The length of a shelf is 28 inches, and the width is 12 inches. How many square feet
of paper is needed to cover the shelf ?
10. Jan wants to carpet a room that measures 15 feet by 12 feet. Carpet is sold at $10.00
per square yard. What will it cost Jan to carpet the room?
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