Page 1 of 8
9.2
Goal
Find the surface areas of
prisms and cylinders.
Key Words
• prism
Surface Area of Prisms
and Cylinders
A prism is a polyhedron with two congruent faces that lie in
parallel planes.
To visualize the surface area of a prism, imagine unfolding it so
that it lies flat. The flat representation of the faces is called a net.
Prism
Unfold
Net
• surface area
• lateral face
• lateral area
• cylinder
The surface area of a polyhedron is the sum of the areas of its faces.
The surface area of a prism is equal to the area of its net.
EXAMPLE
1
Use the Net of a Prism
Find the surface area
of the rectangular
prism.
3
8 in.
5
3 in.
8
5 in.
8
5
3
5
Solution
Add the areas of all the rectangles that form the faces of the prism.
Congruent Faces
Dimensions
Area of Face
Student Help
Left face and right face
8 in. by 5 in.
8 3 5 5 40 in.2
STUDY TIP
Front face and back face
8 in. by 3 in.
8 3 3 5 24 in.2
Top face and bottom face
3 in. by 5 in.
3 3 5 5 15 in.2
Remember that area
is measured in square
units, such as ft 2, m2,
and in.2
Add the areas of all the faces to get the surface area.
S 5 401 40 1 24 1 24 1 15 1 15
5 158
ANSWER
Add the area of all six faces.
Simplify.
© The surface area of the prism is 158 square inches.
9.2
Surface Area of Prisms and Cylinders
483
Page 2 of 8
base
Lateral Faces and Area The lateral faces
of a prism are the faces of the prism that are
not bases. Lateral area is the sum of the
areas of the lateral faces.
lateral
face
base
Surface Area of a Prism One way to find the surface area of
a prism is to use the method in Activity 9.2, summarized below.
Surface area
5
2(area of base)
1
area of lateral faces
5
2B
1
Ph
B
l
w
l
w
h
w
l
Student Help
h
1
5
B w
l
P 5 2l 1 2w
SURFACE AREA OF A PRISM
STUDY TIP
The formula at the right
works for any prism,
regardless of the
shape of its bases.
Words
Surface area 5
2(area of base) 1 (perimeter of base)(height)
Symbols
S 5 2B 1 Ph
EXAMPLE
2
h
l
w
Find Surface Area of a Prism
Find the surface area
of the prism.
5m
3m
4m
Solution
2m
1 Find the area of a triangular base.
●
1
2
Visualize It!
B 5 }} p 4 p 3 5 2 p 3 5 6
It may help you to
visualize the bases and
the height if you redraw
the solid in Example 2
so that it lies on a base.
5
2
3
4
2 Find the perimeter of a base.
●
P 5 3 1 4 1 55 12
3 Find the height of the prism. In the diagram, h 5 2.
●
4 Use the formula for surface area of a prism.
●
S 5 2B 1 Ph
5 2 p 6 1 12 p 2
5 12 1 24
5 36
ANSWER
484
Chapter 9
Formula for the surface area of a prism
Substitute 6 for B, 12 for P, and 2 for h.
Multiply.
Add.
© The surface area of the prism is 36 square meters.
Surface Area and Volume
Page 3 of 8
Find Surface Area of Prisms
Find the surface area of the prism.
1.
2.
3.
8 cm
6 cm
6 ft
3 in.
4 cm
8 ft
6 in.
10 cm
5 ft
2 in.
Surface Area of a Cylinder A cylinder is a solid with two congruent
circular bases that lie in parallel planes. The lateral area of a cylinder is
the area of the curved surface.
base
radius r
The radius of
the cylinder is
the radius of a
base.
height h
lateral
surface
The height of
the cylinder is the
perpendicular
distance between
the bases.
base
Visualize It!
The label below
represents the lateral
surface of a cylinder.
When unwrapped, the
label is a rectangle.
The diagram below shows how to find the surface area of a cylinder.
Surface area
5
5
2(area of base)
2πr
2
B 5 πr 2
1
lateral area
1
(2πr )h
r
r
h
1
5
r
C 5 2πr
r
SURFACE AREA OF A CYLINDER
Words
Surface area 5
2(area of base) 1 (circumference of base)(height)
Symbols
S 5 2B 1 Ch
5 2πr 2 1 2πrh
B 5 πr 2
C 5 2πr
r
9.2
Surface Area of Prisms and Cylinders
485
Page 4 of 8
Student Help
LOOK BACK
See p. 452 to review
calculations with π.
EXAMPLE
3
Find Surface Area of a Cylinder
Find the surface area of the cylinder. Round
your answer to the nearest whole number.
Solution
3 ft
The radius of the base is 3 feet and the height
4 feet. Use these values in the formula for
surface area of a cylinder.
S 5 2πr 2 1 2πrh
Write the formula for surface area.
2
5 2π(3 ) 1 2π(3)(4)
Substitute 3 for r and 4 for h.
5 18π 1 24π
Simplify.
5 42π
Add.
≈ 132
Multiply.
ANSWER
IStudent Help
© The surface area is about 132 square feet.
EXAMPLE
4
Find Lateral Area
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
About how much plastic is used to make a straw that has a diameter
of 5 millimeters and a height of 195 millimeters?
Solution
The straw is a cylinder with no bases. Use the formula for the
surface area of a cylinder, but do not include the areas of the bases.
The diameter is 5 millimeters. So the radius is 5 4 2 5 2.5.
Lateral area 5 2πrh
ANSWER
Surface area formula without bases.
5 2π(2.5)(195)
Substitute 2.5 for r and 195 for h.
5 975π
Simplify.
≈ 3063
Multiply.
© The straw is made with about 3063 square millimeters
of plastic.
Find Surface Area of Cylinders
Find the area described. Round your answer to the nearest
whole number.
4. surface area
3 in.
5. surface area
1m
12 ft
5 in.
10 ft
486
Chapter 9
Surface Area and Volume
6. lateral area
Page 5 of 8
9.2 Exercises
Guided Practice
Vocabulary Check
Tell whether the statement is true or false.
1. The solid is a triangular prism.
2. The blue face is a lateral face of the solid.
3. The red face is a lateral face of the solid.
4. The blue face is a base of the solid.
Skill Check
Find the surface area of the solid. If necessary, round your answer to
the nearest whole number.
5.
6.
7. 4 m
10 in.
6 in.
2m
4 in.
3m
4 ft
Practice and Applications
Extra Practice
See p. 691.
Identifying Parts of a Prism In Exercises 8–10, use the diagram at
the right.
8. What is the height of the solid?
5 in.
9. What is the area of a base?
10. What is the perimeter of a base?
3 in.
2 in.
Identifying Parts of a Cylinder In Exercises 11–13, use the diagram
at the right.
11. What is the height of the solid?
4m
5m
12. What is the area of a base?
Homework Help
Example 1: Exs. 17–20,
28, 29, 31
Example 2: Exs. 17–20,
28, 29, 31
Example 3: Exs. 24–27,
30
Example 4: Exs. 39–44
13. What is the circumference of a base?
Identifying Parts of a Prism In Exercises 14–16, use the diagram at
the right.
10 ft
14. What is the height of the solid?
15. What is the area of a base?
16. What is the perimeter of a base?
6 ft
15 ft
8 ft
9.2
Surface Area of Prisms and Cylinders
487
Page 6 of 8
Surface Area of a Prism Find the surface area of the prism.
17.
18.
19.
6 ft
10 m
4 in.
7m
9 ft
10 ft
4 in.
8m
4 in.
20.
6m
You be the Judge
The height of Prism A is twice
the height of Prism B. Is the
surface area of Prism A twice
the surface area of Prism B?
Explain.
Prism A
Prism B
1m
2m
3m
3m
3m
3m
Analyzing Nets Name the solid that can be folded from the net.
21.
22.
23.
Finding Surface Area of a Cylinder Find the surface area of the
cylinder. Round your answer to the nearest whole number.
24.
25.
6m
26.
8 cm
13 m
8 cm
6 ft
Finding Surface Area Find the surface area of the solid. Round your
answer to the nearest whole number.
2.7 in.
5 in.
27.
28. Cube with
29.
16 mm sides
7 in.
18 in.
30.
31.
3 in.
10 cm
B ≈ 260 cm2
1 in.
11 cm
B ≈ 3 in.2
488
Chapter 9
Surface Area and Volume
Page 7 of 8
32. Error Analysis Juanita is trying to find the surface area of the
cylinder shown below. What did she do wrong?
S 5 2π(122) 1 2π(12)(10)
5 2π(24) 1 2π(120)
5 288π
≈ 905 in.2
12 in.
10 in.
Visualize It! Sketch the solid that results after the net has been
folded. Use the shaded faces as bases.
33.
34.
Visualize It!
35.
Sketch the solid described and find its surface area.
36. A rectangular prism with a height of 10 feet, a length of 3 feet,
and a width of 6 feet.
37. A cylinder with a radius of 3 meters and a height of 7 meters.
38. A triangular prism with a height of 5 inches and a base that is a
right triangle with legs of 8 inches and 6 inches.
Finding Lateral Area Find the lateral area of the solid. If necessary,
round your answer to the nearest whole number.
39.
40.
Music
7m
41.
3 yd
6 ft
8 yd
2m
6m
8 ft
5 ft
42. Clothes Rod Find the lateral area of a clothes rod that has a
radius of 2 centimeters and a height of 90 centimeters.
WAX CYLINDER
RECORDS The first type of
phonograph records were
hollow wax cylinders. The
cylinder is rotated on a
phonograph to produce
the music.
43. Wax Cylinder Records In the late 1800’s, a standard-sized
cylinder record was about 2 inches in diameter and 4 inches
long. Find the lateral area of the cylinder described.
44. Compact Discs A standard compact disc has an outer radius of
60 millimeters and a height of 1.2 millimeters. Find the lateral area
of the disc.
9.2
Surface Area of Prisms and Cylinders
489
Page 8 of 8
Architecture In Exercises 45 and 46 use the following information.
Suppose a skyscraper is a prism that is 415 meters tall and each base
is a square that measures 64 meters on a side.
45. What is the lateral area of this skyscraper?
46. Challenge What is the surface area of this skyscraper? (Hint: The
ground is not part of the surface area of the skyscraper.)
Standardized Test
Practice
47. Multiple Choice What is the approximate surface area of the
cylinder shown?
A
X
C
X
502 cm2
785 cm
2
2
B 628 cm
X
2
D 1570 cm
X
10 cm
15 cm
48. Multiple Choice What is the height of the prism shown if the
surface area is 104 square feet?
F
X
H
X
Mixed Review
4 feet
6 feet
G 5 feet
X
J 7 feet
X
x
5 ft
2 ft
Finding Area Find the area of the polygon made up of rectangles
and triangles. (Lessons 8.3, 8.4)
49.
50.
51. 5 in.
5 cm
6 ft
13 in.
4 cm
11 ft
10 cm
3 in.
52. A parallelogram has a height of 5 feet and an area of 70 square feet.
Find the base. (Lesson 8.5)
Algebra Skills
Solving Proportions Solve the proportion. (Skills Review, p. 660)
53. }} 5 }}
6
x
2
5
54. }} 5 }}
1
10
4
x
55. }} 5 }}
4
7
16
x
57. }} 5 }}
33
x
11
13
58. }} 5 }}
56. }} 5 }}
490
Chapter 9
Surface Area and Volume
3
5
x
35
x
32
5
8