Name
Class
Date
17.2 Surface Area of Prisms
and Cylinders
Essential Question: How can you find the surface area of a prism or cylinder?
Resource
Locker
G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures,
including prisms, ... cylinders, ... to solve problems using appropriate units of measure.
Also G.10.B
Explore
Developing a Surface Area Formula
Surface area is the total area of all the faces and curved surfaces of a three-dimensional figure. The lateral area of a
prism is the sum of the areas of the lateral faces.

Consider the right prism shown here and the net for the right prism. Complete the figure by
labeling the dimensions of the net.
h
a
c
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b
B
In the net, what type of figure is formed by the lateral faces of the prism?

Write an expression for the length of the base of the rectangle.

How is the base of the rectangle related to the perimeter of the base of the prism?

The lateral area L of the prism is the area of the rectangle. Write a formula for L in terms of
h, a, b, and c.
Module 17
1021
Lesson 2
F
Write the formula for L in terms of P, where P is the perimeter of the base of the prism.
GLet B be the area of the base of the prism. Write a formula for the surface area S of the prism
in terms of B and L. Then write the formula in terms of B, P, and h.
Reflect
1.
Explain why the net of the lateral surface of any right prism will always be a rectangle.
2.
Suppose a rectangular prism has length ℓ, width w, and height h, as shown. Explain how you can write a
formula for the surface area of the prism in terms of ℓ, w, and h.
h
ℓ
w
Explain 1 Finding the Surface Area of a Prism
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Lateral Area and Surface Area of Right Prisms
The lateral area of a right prism with height h and base perimeter P is L = Ph.
The surface area of a right prism with lateral area L and base area B is S = L + 2B,
or S = Ph + 2B.
h
B
Module 17
1022
Lesson 2
Example 1
Each gift box is a right prism. Find the total
amount of paper needed to wrap each box,
not counting overlap.

Step 1 Find the lateral area.
Lateral area formula
L = Ph
P = 2(8) + 2(6) = 28 cm
= 28(12)
Multiply.
= 336 cm 2
12 cm
6 cm
Step 2 Find the surface area.
Surface area formula
8 cm
S = L + 2B
Substitute the lateral area.
= 336 + 2(6)(8)
Simplify.
= 432 cm 2
B
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Studios/Photodisc/Getty Images
Step 1 Find the length c of the hypotenuse of the base.
Pythagorean Theorem
c =a +b
2
2
Substitute.
=
Simplify.
=
Take the square root of each side.
2
10 in.
2
24 in.
2
+
20 in.
c=
Step 2 Find the lateral area.
Lateral area formula
Module 17
L = Ph
Substitute.
=
Multiply.
=
1023
(
)
in 2
Lesson 2
Step 3 Find the surface area.
S = L + 2B
Surface area formula
Substitute.
=
Simplify.
=
1  ​ 
+ 2 ⋅ ​ _
2
2
i​n ​ ​
⋅
Reflect
3.
A gift box is a rectangular prism with length 9.8 cm, width 10.2 cm, and height 9.7 cm. Explain how to
estimate the amount of paper needed to wrap the box, not counting overlap.
Your Turn
Each gift box is a right prism. Find the total amount of paper needed to wrap each box,
not counting overlap.
4.
5.
5 in.
18 in.
6 in.
5 in.
3.6 in.
8.5 in.
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Module 17
1024
Lesson 2
Explain 2
Finding the Surface Area of a Cylinder
Lateral Area and Surface Area of Right Cylinders
The lateral area of a cylinder is the area of the curved surface that connects the two bases.
The lateral area of a right cylinder with radius r and height h is L = 2πrh.
The surface area of a right cylinder with lateral area L and base area B is
S = L + 2B, or S = 2πrh + 2πr 2.
r
r
2πr
h
h
Example 2
Each aluminum can is a right cylinder. Find the amount of paper needed
for the can’s label and the total amount of aluminum needed to make the
can. Round to the nearest tenth.

3 cm
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Step 1 Find the lateral area.
Lateral area formula
L = 2πrh
Substitute.
L = 2π(3)(9)
9 cm
= 54π cm 2
Multiply.
Step 2 Find the surface area.
Surface area formula
S = L + 2πr 2
Substitute the lateral area and radius.
= 54π + 2r(3)
Simplify.
= 72π cm 2
2
Step 3 Use a calculator and round to the nearest tenth.
The amount of paper needed for the label is the lateral area, 54π ≈ 169.6 cm 2.
The amount of aluminum needed for the can is the surface area, 72π ≈ 226.2 cm 2.
Module 17
1025
Lesson 2
5 in
B
Step 1 Find the lateral area.
Lateral area formula
Substitute; the radius is half the diameter.
Step 2 Find the surface area.
Surface area formula
Substitute the lateral area and radius.
( )​​( )​
= 2π​
=
Multiply.
π ​in ​2​
S = L + 2π​r ​2​
Simplify.
2 in
L = 2πrh
( )​ ​ ​
=
π + 2r​​
=
π ​in ​2​
2
Step 3 Use a calculator and round to the nearest tenth.
π≈
The amount of paper needed for the label is the lateral area,
The amount of aluminum needed for the can is the surface area,
i​n ​2​.
π≈
i​n ​2​.
Reflect
6.
In these problems, why is it best to round only in the final step of the solution?
Your Turn
Each aluminum can is a right cylinder. Find the amount of paper needed for the can’s
label and the total amount of aluminum needed to make the can. Round to the
nearest tenth.
7.
8.
80 mm
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72 mm
15 cm
6 cm
Module 17
1026
Lesson 2
Explain 3
Example 3
Finding the Surface Area of a Composite Figure
Find the surface area of each composite figure. Round to the nearest tenth.
4 ft

Step 1 Find the surface area of the right rectangular prism.
Surface area formula
S = Ph + 2B
Substitute.
= 80(20) + 2(24)(16)
Simplify.
= 2368 ft 2
20 ft
16 ft
24 ft
Step 2 A cylinder is removed from the prism. Find the lateral
area of the cylinder and the area of its bases.
Lateral area formula
L = 2πrh
Substitute.
= 2π(4)(20)
Simplify.
= 160π ft 2
Base area formula
B = πr 2
Substitute.
= π(4)
Simplify.
= 16π ft 2
2
Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of
all surfaces on the exterior of the figure.
S = (prism surface area) + (cylinder lateral area) - (cylinder base areas)
= 2368 + 160π - 2(16π)
= 2368 + 128π
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≈ 2770.1 ft 2
Module 17
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Lesson 2
B
2 cm
3 cm
Step 1 Find the surface area of the right rectangular prism.
Surface area formula
S = Ph + 2B
Substitute.
=
Simplify.
=
( ) + 2( )( )
cm 2
5 cm
4 cm
9 cm
Step 2 Find the surface area of the cylinder.
Lateral area formula
L = 2πrh
( )( )
Substitute.
= 2π
Simplify.
=
Surface area formula
π cm 2
S = L + 2πr 2
( )
Substitute.
=
π + 2π
Simplify.
=
π cm 2
2
Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of all
surfaces on the exterior of the figure.
S = (prism surface area) + (cylinder surface area) - 2(area of one cylinder base)
(
=
+
π - 2π
=
+
π≈
)
2
cm 2
Reflect
9.
Discussion A student said the answer in Part A must be incorrect since a part of the rectangular prism
is removed, yet the surface area of the composite figure is greater than the surface area of the rectangular
prism. Do you agree with the student? Explain.
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Module 17
1028
Lesson 2
Your Turn
Find the surface area of each composite figure. Round to the nearest tenth.
10.
11.
5 in
7 mm
3 in
6 mm
3 in 5 in
7 in
3 mm
9 in
Elaborate
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12. Can the surface area of a cylinder ever be less than the lateral area of the cylinder? Explain.
13. Is it possible to find the surface area of a cylinder if you know the height and the circumference of the
base? Explain.
14. Essential Question Check-In How is finding the surface area of a right prism similar to finding the
surface area of a right cylinder?
Module 17
1029
Lesson 2
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Find the lateral area and surface area of each prism.
1.
2.
4 cm
3 cm
2 cm
3 ft
7 ft
5 cm
5 ft
3.
4.
15 cm
10 cm
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12 m
10.39 m
5 cm
5 cm
Module 17
1030
Lesson 2
Find the lateral area and surface area of the cylinder. Leave your answer in terms of π.
5.
6.
3 ft
11 in.
4 ft
7 in.
Find the total surface area of the composite figure. Round to the nearest tenth.
7.
8.
4 ft
6 ft
14 ft
14 ft
8 ft
14 ft
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12 ft
8 ft
14 ft
Module 17
1031
Lesson 2
Find the total surface area of the composite figure. Round to the
nearest tenth.
9.
10.
2 ft
8 cm
2 ft
0.5 ft
2 cm
6 cm
2 ft
9 cm
10 cm
1 ft
11. The greater the lateral area of a florescent light bulb, the more light the bulb produces.
One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical
bulb is 23 inches long with a __
​ 34 ​-inch radius. Which bulb will produce more light?
Module 17
13. Find the lateral and surface area of a cylinder with
base area 64π ​m ​2​and a height 3 meters less
than the radius.
1032
Lesson 2
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12. Find the lateral and surface area of a cube with edge
length 9 inches.
14. Biology Plant cells are shaped approximately
like a right rectangular prism. Each cell absorbs
oxygen and nutrients through its surface. Which
cell can be expected to absorb at a greater rate?
(Hint: 1 μm = 1 micrometer = 0.000001 meter)
15 µm
7 µm
10 µm
35 µm
11 µm
15 µm
ge07se_c10l04003a
AB
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15. Find the height of a right cylinder with surface area
160π f​t ​2​and radius 5 ft.
16. Find the height of a right rectangular prism with
surface area 286 m
​  ​2​, length 10 m, and width 8 m.
17. Represent Real-World Problems If one gallon of paint covers
250 square feet, how many gallons of paint will be needed to cover
the shed, not including the roof? If a gallon of paint costs $25,
about how much will it cost to paint the walls of the shed?
12 ft
18 ft
12 ft
Module 17
1033
18 ft
Lesson 2
18. Match the Surface Area with the appropriate coin in the table.
Coin
Diameter (mm)
Thickness (mm)
Penny
19.05
1.55
Nickel
21.21
1.95
Dime
17.91
1.35
Quarter
24.26
1.75
Surface Area (m​m ​2​)
A.836.58
B.579.82
C.662.81
D.1057.86
19. Algebra The lateral area of a right rectangular prism is 144 c​m ​2​. Its length is three
times its width, and its height is twice its width. Find its surface area.
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20. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another
cylinder that has a radius of 4 cm and the same surface area as the first cylinder.
Module 17
1034
Lesson 2
H.O.T. Focus on Higher Order Thinking
10
f
t
21. Analyze Relationships Ingrid is building a shelter to protect her plants from
freezing. She is planning to stretch plastic sheeting over the top and the ends of the
frame. Assume that the triangles in the frame on the left are equilateral. Which of the
frames shown will require more plastic? Explain how finding the surface area of these
figures is different from finding the lateral surface area of a figure.
10
10
ft
ft
10 ft
10 ft
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ge07sec10l04004aa
1st pass
4/23/5
cmurphy
ge07sec10l04005a
1st pass
4/12/5
cmurphy
22. Communicate Mathematical Ideas Explain how to use the net of a
three-dimensional figure to find its surface area.
23. Draw Conclusions Explain how the edge lengths of a rectangular prism
can be changed so that the surface area is multiplied by 9.
Module 17
1035
Lesson 2
Lesson Performance Task
A manufacturer of number cubes has the bright idea of
packaging them individually in cylindrical boxes. Each number
cube measures 2 inches on a side.
1. What is the surface area of each cube?
2. What is the surface area of the cylindrical box?
Assume the cube fits snugly in the box and that the
box includes a top. Use 3.14 for π.
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Module 17
1036
Lesson 2