Lesson 37
Using Formulas to Find
Surface Areas and Volumes
TAKS Grade 9 Objective 8
(8.8)(C)
The volume of a solid is the measure of the space that the solid occupies. In other
words, the volume is the number of cubic units that is needed to fill the solid.
The volumes and surface areas of many solids can be found using formulas.
To solve a problem using a formula, substitute the values into the formula
and evaluate.
New Vocabulary
• volume
Volume Formulas
The volume of a prism or a cylinder can be found by taking the product of
the area of one base and the height. Let B represent the area of one base
and h represent the height.
V Bh
The volume of a pyramid or a cone can be found by taking one-third the
product of the area of the base and the height. Let B represent the area of
the base and h represent the height.
V 13Bh
The volume of a sphere with a radius of r can be determined with the
following formula.
V 34r3
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EXAMPLE 1
• The height of a
prism or cylinder
is the length of a
perpendicular line
segment connecting
the bases.
• The height of a
pyramid or cone is
the length of the
perpendicular line
segment connecting
the base and the
vertex.
Find the approximate volume of the cone.
Step 1 Find the area of the base.
A r2 (3.14)(8 ft)2 200.96 ft2
The area of the base of the cone is approximately 201 square feet.
Step 2 Use the area of the base and the height of the cone to
find the volume.
V 13Bh 13(201 ft2)(15 ft) 1005 ft3
15 ft
8 ft
The volume of the cone is approximately 1005 cubic feet.
Quick Check 1
1a. Find the volume of the solid.
4 cm
1b. Find the volume of the solid, and round the
answer to the nearest cubic centimeter.
5 cm
3 cm
12 cm
TAKS Review and Preparation Workbook
14 cm
LESSON 37
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Using Formulas to Find Surface Areas and Volumes
109
TAKS Objective 8 (8.8)(C)
LESSON 37
Lateral Surface Area and Surface Area Formulas
The lateral surface area of a prism can be found by taking the product of the perimeter of the base and
the height of the prism.
LA ph
The lateral surface area of a cylinder with a radius of r can be found by taking the product of the
circumference of the base and the height of the cylinder.
LA 2rh
The surface area of a prism or a cylinder can be found by taking the sum of the lateral surface area and
the areas of its bases.
SA LA 2B
The lateral surface area of a square pyramid can be found by taking 4 times the area of one lateral
triangular face.
LA 4 21bl 2bl
The lateral surface area of a cone with radius r and slant height l can be found by taking 12 the product
of the circumference of the base and the slant height.
LA 12 (2r)l rl
The surface area of a square pyramid or cone can be found by taking the sum of the lateral surface area
and the area of its base.
SA LA B
The surface area of a sphere with a radius of r can be determined with the following formula.
SA 6s2
EXAMPLE 2
Find the approximate surface area of the cone.
Step 1 Find the lateral surface area of the cone.
LA rl (3.14)(10 in.)(15 in.) 471 in.2
Step 2 Find the area of the base of the cone.
B r2 (3.14)(10 in.)2 314 in.2
Step 3 Find the surface area by taking the sum of the
lateral surface area and the area of the base.
SA LA B 471 in.2 314 in.2 785 in.2
The surface area of the cone is approximately 785 square inches.
15 in.
20 in.
Quick Check 2
2a. A square pyramid has a base area of 121 m2 and a slant height of 20 m.
Find the lateral surface area and surface area of the solid. Round answers
to the nearest square meter.
2b. A cylinder has a base circumference of 6 meters and a height of 2
meters. Find the lateral surface area and surface area of the solid. Round
answers to the nearest square meter.
110
LESSON 37
■
Using Formulas to Find Surface Areas and Volumes
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
SA 4r2
The surface area of a cube with an edge of length s can be determined with the following formula.
Name__________________________Class____________Date________
1 A cylindrical wastebasket has a radius of 5
inches. If the volume is 1884 cubic inches,
which of the following is the best estimate
for the height of the wastebasket?
4 A paper drinking cup is in the shape of a
square pyramid.
3 in.
A 12 inches
5 in.
B 24 inches
C 30 inches
D 60 inches
How much paper is needed to make 10 of
these paper cups?
2 The rental cost of a storage unit at Joe’s
Storage is $0.75 per cubic foot. What is the
rental cost of the storage unit with the
dimensions shown below?
F 150 square inches
G 300 square inches
H 390 square inches
J 600 square inches
8 ft
5 The following aquarium is built to fit in the
corner of a room. How much water is needed
to fill the tank halfway?
10 ft
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
6 ft
F $192
0.6 m
G $360
0.8 m
H $480
J $640
2m
3 Estimate the total surface area of the cone
shown below.
1m
A 0.24 cubic meters
B 0.48 cubic meters
5 yd
C 2.4 cubic meters
4 yd
D 4.8 cubic meters
6 yd
A 38 yd2
C 66 yd2
B 47 yd2
D 75 yd2
TAKS Review and Preparation Workbook
6 Which best represents the surface area of a
kickball that has a radius of 1 foot?
LESSON 37
■
F p square feet
G 2p square feet
H 43p square feet
J 4p square feet
Using Formulas to Find Surface Areas and Volumes
111