LESSON
17.2
Name
Surface Area of Prisms
and Cylinders
Class
Date
17.2 Surface Area of Prisms
and Cylinders
Essential Question: How can you find the surface area of a prism or cylinder?
Texas Math Standards
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
The student is expected to:
G.11.C
Explore
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
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.

Mathematical Processes
Consider the right prism shown here and the net for the right prism. Complete the figure by
labeling the dimensions of the net.
G.1.F
Analyze mathematical relationships to connect and communicate
mathematical ideas.
Language Objective
h
h
a
1.B, 2.E.3, 3.E, 3.H.3, 4.D
b
Explain to a partner how to find the surface area of prisms and cylinders.
a
ENGAGE
You find the lateral area and then add twice the
area of a base.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the
photograph. Ask students to identify the subject of
the photo and to speculate on the significance of the
surface area on determining how items are packaged.
Then preview the Lesson Performance Task.

c
In the net, what type of figure is formed by the lateral faces of the prism?
rectangle
© Houghton Mifflin Harcourt Publishing Company
Essential Question: How can you find
the surface area of a prism or a cylinder?
b
c

Write an expression for the length of the base of the rectangle.
a+b+c

How is the base of the rectangle related to the perimeter of the base of the prism?
They are equal.

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.
L = h(a + b + c)
Module 17
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Lesson 2
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GE_MTXESE353893_U7M17L2 1021
HARDCOVER PAGES 823832
Turn to these pages to
find this lesson in the
hardcover student
edition.
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prism shown
the right
of the net.
Consider
dimensions
labeling the

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Lesson 2
1021
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GE_MTX
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Lesson 17.2
1021
2/22/14
2:12 AM
20/01/15 5:53 PM
F
G
Write the formula for L in terms of P, where P is the perimeter of the base of the prism.
L = Ph
EXPLORE
Let 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.
Developing a Surface Area Formula
S = L + 2B; S = Ph + 2B
INTEGRATE TECHNOLOGY
Reflect
1.
Students have the option of doing the Explore activity
either in the book or online.
Explain why the net of the lateral surface of any right prism will always be a rectangle.
Sample answer: Each lateral face of any right prism is a rectangle. The net of the lateral
surface of any right prism is composed of rectangles joined end-to-end. Straight angles
are formed when the rectangles are joined in this manner resulting in one long
QUESTIONING STRATEGIES
rectangular shape.
2.
In a prism, how is the lateral area formula
related to the surface area formula? The
surface area formula consists of the lateral area plus
the area of the bases.
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
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Sample answer: There are two faces with area ℓw, two faces with area wh, and two faces
with area ℓh, so the surface area can be written as S = 2ℓw + 2wh + 2ℓh.
Explain 1
Have students brainstorm how to determine what
three-dimensional figure can be made from a given
net and how the net can be used to find the surface
area of the figure. Emphasize that prisms have
parallelograms for sides, and cylinders have
congruent circular bases.
Finding the Surface Area of a Prism
© Houghton Mifflin Harcourt Publishing Company
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
EXPLAIN 1
Finding the Surface Area of a Prism
QUESTIONING STRATEGIES
Module 17
1022
Lesson 2
PROFESSIONAL DEVELOPMENT
GE_MTXESE353893_U7M17L2.indd 1022
Integrate Mathematical Processes
This lesson provides an opportunity to address TEKS G.1.F, which calls for
students to “analyze relationships.” In this lesson, students analyze
three-dimensional figures to determine how they “decompose” into twodimensional faces, each with its own area, and to find that the sum of the areas of
the faces is equal to the surface area of the figure. Since the faces of the figures are
polygons or circles, the combined areas generate the lateral area and surface area
formulas students will use in this lesson.
2/22/14 2:12 AM
How can you use the formula for the area of a
parallelogram to find the lateral area of a
prism? Because the lateral faces of a prism are
parallelograms, you can use the parallelogram
formula to find the areas of the lateral faces and
then add them together.
Surface Area of Prisms and Cylinders 1022
Example 1
QUESTIONING STRATEGIES
When can the Pythagorean Theorem be used
to find the area of the bases of a triangular
prism? If the bases are right triangles, then the
Pythagorean Theorem can be used to find the
lengths of the legs of the triangles, which are
necessary to find the area of the triangles.
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
12 cm
2
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

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©C Squared
Studios/Photodisc/Getty Images
Step 1 Find the length c of the hypotenuse of the base.
Pythagorean Theorem
10 in.
c2 = a2 + b2
2
Substitute.
= 10 + 24
Simplify.
= 676
Take the square root of each side.
24 in.
2
20 in.
c = 26
Step 2 Find the lateral area.
Lateral area formula
L = Ph
Substitute.
= 60
Multiply.
=
Module 17
( 20 )
1200
1023
in 2
Lesson 2
COLLABORATIVE LEARNING
GE_MTXESE353893_U7M17L2.indd 1023
Small Group Activity
Have students work in groups to find the surface areas of various prisms and
cylinders. Have students each choose a prism or a cylinder and conjecture how to
find the surface area. Then have them draw and label a model or a net and
describe how to find the surface area. Ask them to verify or disprove their
conjectures, and present their results to the group.
1023
Lesson 17.2
1/21/15 7:31 PM
Step 3 Find the surface area.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Patterns
S = L + 2B
Surface area formula
Substitute.
=
1200
Simplify.
=
1440
1
+2⋅ _
24 ⋅ 10
2
in 2
Encourage students to make an organized list of the
dimensions of the lateral sides and the bases of a
prism as part of their plan for finding the surface
area. Then have them substitute the appropriate
values into the formulas for lateral area and surface
area of a prism.
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.
Sample answer: Round each dimension to 10 cm. Then each face has an area of
approximately 10 2 = 100 cm 2, and the surface area is approximately 6(100) = 600 cm 2.
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.
The lateral area is L = Ph.
Let b be the unknown length
of the leg of the base.
So, L = 46(5) = 230 in 2.
By the Pythagorean Theorem, c 2 = a 2 + b 2,
P = 2(18) + 2(5) = 46 in.
so 6 2 = 3.6 2 + b 2, 36 = 12.96 + b 2,
and b 2 = 23.04.
The surface area is S = L + 2B.
B = 18(5) = 90 in
2
Taking the square root of each side
shows that b = 4.8 in.
So, S = 230 + 2(90) = 410 in 2.
© Houghton Mifflin Harcourt Publishing Company
The lateral area is L = Ph.
P = 3.6 + 4.8 + 6 = 14.4 in.
So, L = 14.4(8.5) = 122.4 in 2.
The surface area is S = L + 2B.
1(
B = __
4.8)(3.6) = 8.64
2
So, S = 122.4 + 2(8.64) = 139.68 in 2.
Module 17
1024
Lesson 2
DIFFERENTIATE INSTRUCTION
GE_MTXESE353893_U7M17L2.indd 1024
1/21/15 7:31 PM
Multiple Representations
Have students work in groups to cover boxes and cylinders with wrapping paper.
Ask them to cut the wrap so that it does not overlap, and have them decompose
the wraps into nets that they can use to find the surface area. Have groups discuss
how the nets are related to the lateral area and the surface area formulas.
Surface Area of Prisms and Cylinders 1024
Explain 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.
Finding the Surface Area of a Cylinder
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.
QUESTIONING STRATEGIES
How is the height of a right cylinder used to
find its surface area? The height is used to
find the lateral area. The lateral area is the
circumference of the base times the height. Adding
the lateral area to the area of the bases gives the
surface area.
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
© Houghton Mifflin Harcourt Publishing Company
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
GE_MTXESE353893_U7M17L2 1025
1025
Lesson 17.2
1025
Lesson 2
22/02/14 5:02 AM
5 in
B
AVOID COMMON ERRORS
Step 1 Find the lateral area.
( )( 2 )
Substitute; the radius is half the diameter.
= 2π 2.5
Multiply.
= 10 π in 2
Common errors students make when applying the
surface area formula include multiplying the height
of the cylinder by the area of the base; using a
diameter in the formula for cylinders instead of a
radius; and forgetting to include the area of both
bases. Caution students to look for these errors.
2 in
L = 2πrh
Lateral area formula
Step 2 Find the surface area.
S = L + 2πr 2
Surface area formula
( )
Substitute the lateral area and radius.
= 10 π + 2r 2.5
Simplify.
= 22.5 π in 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, 10 π ≈ 31.4 in 2.
The amount of aluminum needed for the can is the surface area, 22.5 π ≈ 70.7 in 2.
Reflect
6.
In these problems, why is it best to round only in the final step of the solution?
Sample answer: This results in a more accurate answer. If you round at an intermediate
step, the inaccuracies may be compounded as you perform subsequent operations.
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
15 cm
The radius of the cylinder is half the
diameter, so r = 36 mm.
The lateral area is L = 2πrh.
6 cm
So, L = 2π(36)(80) = 5760π mm 2.
The lateral area is L = 2πrh.
So, L = 2π(6)(15) = 180π cm 2.
The surface area is S = L + 2πr 2.
So, S = 180π + 2π(6) = 252π cm 2.
The amount of paper needed for the label is
the lateral area, 5760π ≈ 18,095.6 mm 2.
So, S = 5760π + 2π(36) = 8352π mm 2.
2
The surface area is S = L + 2πr 2.
2
The amount of paper needed for the label is
the lateral area, 180π ≈ 565.5 cm 2.
The amount of aluminum needed for the can
is the surface area, 252π ≈ 791.7 cm 2.
Module 17
GE_MTXESE353893_U7M17L2 1026
1026
© Houghton Mifflin Harcourt Publishing Company
72 mm
The amount of aluminum needed for the can
is the surface area, 8352π ≈ 26,238.6 mm 2.
Lesson 2
20/01/15 5:53 PM
Surface Area of Prisms and Cylinders 1026
Explain 3
EXPLAIN 3
Example 3
Finding the Surface Area of a
Composite Figure
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
QUESTIONING STRATEGIES
Is the surface area of a composite figure always
equal to the sum of the areas of the parts of
the figure? Explain. No; you must subtract the areas
of any parts of the surface that are overlapping.
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.
2
= π(4)
Simplify.
= 16π ft 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
© Houghton Mifflin Harcourt Publishing Company
≈ 2770.1 ft 2
Module 17
GE_MTXESE353893_U7M17L2 1027
1027
Lesson 17.2
1027
Lesson 2
20/01/15 5:53 PM

2 cm
3 cm
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Patterns
Step 1 Find the surface area of the right rectangular prism.
Surface area formula
S = Ph + 2B
( 5 ) + 2( 9 )( 4 )
Substitute.
= 26
Simplify.
= 202 cm 2
5 cm
Encourage students to carefully decompose a figure
as part of their plan to find its surface area. Have
them make an organized list of the dimensions of the
lateral sides and of the bases for each figure, along
with a list of those areas that are overlapping in the
composite figure. Then have them write an equation
for the total surface area of the parts, including
subtractions for overlapping parts, and substitute the
appropriate values into the formulas.
4 cm
9 cm
Step 2 Find the surface area of the cylinder.
Lateral area formula
L = 2πrh
( 2 )( 3 )
Substitute.
= 2π
Simplify.
= 12 π cm 2
Surface area formula
S = L + 2πr 2
(2)
Substitute.
= 12 π + 2π
Simplify.
= 20 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)
(
= 202 + 20 π - 2π
= 202 + 12 π ≈
2
239.7
)
2
cm 2
Reflect
9.
creates additional exposed area on the interior surface of the hole.
Module 17
GE_MTXESE353893_U7M17L2 1028
1028
© Houghton Mifflin Harcourt Publishing Company
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.
No; removing part of the rectangular prism produces a hole through the prism and this
Lesson 2
20/01/15 5:53 PM
Surface Area of Prisms and Cylinders 1028
Your Turn
ELABORATE
Find the surface area of each composite figure. Round to the nearest tenth.
10.
11.
5 in
QUESTIONING STRATEGIES
7 mm
3 in
How do you find the surface area of a
prism? You add the perimeter of the base
times the height to twice the area of the base.
6 mm
3 in 5 in
7 in
3 mm
9 in
How do you find the surface area of a
cylinder? You add the circumference of the
base times the height to twice the area of the base.
The surface area of the large cylinder is
S large = 2πrh + 2πr 2.
The surface area of the large prism is
S large = Ph + 2B.
So, S large = (32)(5) + 2(9)(7) = 286 in 2.
So, S large = 2π(7)(6) + 2π(7) = 182π mm 2.
The surface area of the small prism is
S small = Ph + 2B.
So, L small = 2π(3)(6) = 36π mm 2.
2
The lateral area of the small prism is L small = 2πrh.
So, S small = (16)(3) + 2(5)(3) = 78 in 2.
SUMMARIZE THE LESSON
The area of each base of the small cylinder
is B = πr 2 = π3 2 = 9π mm 2.
The surface area of the composite figure is
the surface area of the large prism plus the
surface area of the small prism minus 2 times
the area of the base of the small prism.
What is the same about finding the surface
area of a prism and a cylinder? What is
different? For both a prism and a cylinder, you find
the surface area by finding the lateral area and then
adding twice the area of the base; the bases of
prisms and cylinders are different, so finding the
lateral areas and base areas will require
different processes.
The surface area of the composite figure is the
surface area of the large cylinder plus the lateral
area of the small cylinder minus 2 times the area
of the base of the small cylinder.
S = 286 + 78 - 2(5)(3) = 344 in 2
S = 182π + 36π - 2(9π) = 200π ≈ 628.3 mm 2
Elaborate
12. Can the surface area of a cylinder ever be less than the lateral area of the cylinder? Explain.
No. The surface area is the lateral area plus the area of the two bases. Since the area of the
© Houghton Mifflin Harcourt Publishing Company
two bases is greater than 0, the surface area must be greater than the lateral area.
13. Is it possible to find the surface area of a cylinder if you know the height and the circumference of the
base? Explain.
Yes. You can use the circumference of the base to find the radius of the base. Then you can
use the height, circumference, and radius in the surface area formula.
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?
In both cases, you can find the surface area by finding the lateral area and then adding
twice the area of a base.
Module 17
1029
Lesson 2
LANGUAGE SUPPORT
GE_MTXESE353893_U7M17L2.indd 1029
Connect Vocabulary
To help students remember the vocabulary in the lesson, including lateral area
and surface area, have students make note cards of several different solid figures
and their lateral and surface areas. Then have them use colored pencils to mark
the dimensions of each in one color, and the formulas they will use in another
color. Have them label the figures with the units and show the substitutions for the
formulas. Ask them to share their note cards with other students
1029
Lesson 17.2
1/21/15 7:31 PM
Evaluate: Homework and Practice
EVALUATE
• 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
= (12)2
L = Ph
= 24 cm 2
= (24)3
The base is a 3–4–5 right triangle, so in the
area formula, b = 3 and h = 4.
= 72 ft 2
S = Ph + 2B
S = Ph + 2B
= 72 + 2(5)(7)
(
)
1 ( )( )
= 24 + 2 _
3 4
2
= 72 + 70
= 24 + 12
= 142 ft 2
L = 72 ft
ASSIGNMENT GUIDE
L = Ph
= 36 cm 2
2
L = 24 cm 2
S = 142 ft 2
S = 36 cm 2
3.
4.
Concept and Skills
Practice
Explore
Developing a Surface Area Formula
Exercise 11
Example 1
Finding the Surface Area of a Prism
Exercises 1–4
Example 2
Finding the Surface Area
of a Cylinder
Exercises 5–6
Example 3
Finding the Surface Area of a
Composite Figure
Exercises 7–10
15 cm
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
10 cm
L = Ph
5 cm
= (72)15 = 1080 m 2
5 cm
The base can be divided into twelve right
triangles, each triangle with a height of
10.39 m and a base of 6 m.
L = Ph
= (20)10
= 200 cm
1
B=_
bh(12)
2
2
1 ( )(
=_
6 10.39)(12)
2
S = Ph + 2B
= 374.04 m 2
= 200 + 2(5)(5)
= 1080 + 2(374.04)
= 250 cm 2
≈ 1828.08 m 2
L = 200 cm 2
L = 1080 m 2
S = 250 cm 2
S = 1828.08 m 2
Module 17
Exercise
Depth of Knowledge (D.O.K.)
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
Lesson 2
1030
Mathematical Processes
1 Recall of Information
1.C Select tools
11
2 Skills/Concepts
1.A Everyday life
12–20
2 Skills/Concepts
1.B Problem solving model
21
3 Strategic Thinking
1.A Everyday life
22
3 Strategic Thinking
1.D Multiple representations
23
3 Strategic Thinking
1.F Analyze relationships
1–10
Some students may benefit from a hands-on
approach for finding the surface area of solids. Have
students draw simple figures like prisms and
cylinders and then discuss in groups how they can
find the lateral areas and the surface areas. Have them
include a discussion of the properties of the faces of
the figures that will help them find the lateral areas or
the surface areas.
S = L + 2B
= 200 + 50
GE_MTXESE353893_U7M17L2.indd 1030
© Houghton Mifflin Harcourt Publishing Company
10.39 m
12 m
2/22/14 2:11 AM
Some students may benefit from using the
programming features of a graphing calculator to
find the surface areas of right rectangular prisms and
right cylinders. Have students enter the formulas for
the surface areas of these simple solids as output from
a program, with the dimensions of the
solids as inputs.
Surface Area of Prisms and Cylinders 1030
Find the lateral area and surface area of the cylinder. Leave your answer in terms of π.
AVOID COMMON ERRORS
5.
As students find the surface area of cylinders, caution
them to avoid the common errors of forgetting to
include the areas of both bases, or using the diameter
of the base instead of the radius in the formula.
6.
3 ft
11 in.
4 ft
7 in.
L = 2πrh
L = 2πrh
= 2π(3)(4)
= 2π(5.5)(7)
= 24π ft 2
S = L + 2πr
= 77π in 2
S = L + 2πr 2
2
= 24π + 2π(3)
= 77π + 2π(5.5)
2
= 24π + 18π
= 77π + 60.5π
= 42π ft 2
L = 24π ft
S = 42π ft
2
= 137.5π in 2
L = 77π in 2
2
S = 137.5π in 2
2
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
© Houghton Mifflin Harcourt Publishing Company
14 ft
12 ft
8 ft
14 ft
Surface Area of Cylinder
L = 2πrh
Surface Area of Cylinder
L = 2πrh
= 2π(4)(8)
= 64π ft
S = L + 2πr 2
= 64π + 2π(4)
= 96π ft
2
= 2π(14)(14)
= 392π ft 2
2
L = Ph
= 528 ft 2
= (40)14
S = L + 2B
= 560 ft 2
= 528 + 2(14)(8)
= (44)12
784π + 560 - 2(14 ⋅ 6) ≈ 2855.0 ft 2
= 752 ft 2
S ≈ 2855.0 ft 2
96π - π(4) + 752 - π(4) ≈ 953.1 ft 2
2
= 392π + 2π14 2
= 784π ft 2
Lateral Surface Area of Prism
2
Surface Area of Prism
L = Ph
S = L + 2πr 2
2
S ≈ 953.1 ft 2
Module 17
GE_MTXESE353893_U7M17L2 1031
1031
Lesson 17.2
1031
Lesson 2
2/5/15 5:40 PM
Find the total surface area of the composite figure. Round to the
nearest tenth.
9.
10.
2 ft
8 cm
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
2 ft
0.5 ft
Because a cylinder has circular bases, the
circumference of the bases is the perimeter of the
bases. Therefore, the lateral area of the right cylinder
depends on the circumference of the base. If students
think about the net for a cylinder, the net includes a
rectangle and two circles. That means that the
rectangle must have length equal to the
circumference of the base.
2 cm
6 cm
2 ft
9 cm
10 cm
1 ft
Surface Area of Prism
Surface Area of Prism
The base is a 6–8–10, (3–4–5), right
triangle, so in the area formula
b = 8 and h = 6.
S = L + 2B
L = Ph = (24)9 = 216 cm 2
S = L + 2B
(
L = Ph = (8)0.5 = 4 ft 2
= 4 + 2(2)(2) = 12 ft 2
Total Area of Cylinder
L = 2πrh
)
1 ( )( )
= 216 + 2 _
8 6 = 264 cm 2
2
= 2π(0.5)(2) = 2π ft 2
Lateral Surface Area of Cylinder
S = L + 2πr 2
L = 2πrh
= 2π + 2π0.5 2 = 2.5π ft 2
= 2π(2)(9) = 36π cm 2
12 + 2.5π - 2(π0.5 2) ≈ 18.3 ft 2
264 + 36π - 2(π2 2) ≈ 352.0 cm 2
S ≈ 18.3 ft 2
S ≈ 352.0 cm 2
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?
Lateral Area of 23 inch bulb
L = 2πrh
L = 2πrh
= 2π(1)(16)
= 32π in 2
= 2π(0.75)(23)
= 34.5π in 2
The 23 inch bulb will produce more light.
12. Find the lateral and surface area of a cube with edge
length 9 inches.
L = Ph
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.
L = 2πrh
Find the Radius
= (36)9
A = πr 2
= 324 in 2
64π = πr 2
64π
πr 2
π = π
64 = r 2
S = L + 2B
_ _
= 324 + 2(9)(9)
= 324 + 162
8=r
= 486 in 2
L = 324 in
h=r-3
2
h=8-3
S = 486 in 2
Module 17
GE_MTXESE353893_U7M17L2 1032
1032
h=5
= 2π(8)(5)
= 80π m 2
S = L + 2πr 2
= 80π + 2π(8)
= 208π m
© Houghton Mifflin Harcourt Publishing Company
Lateral Area of 16 inch bulb
2
2
L = 80π m 2
S = 208π m 2
Lesson 2
20/01/15 5:53 PM
Surface Area of Prisms and Cylinders 1032
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
Surface Area of Cell 1
L = Ph
= (90)7
= 630 μm
11 µm
10 µm
15 µm
35 µm
Surface Area of Cell 2
S = L + 2B
2
L = Ph
= 630 + 2(35)(10)
= (52)15
= 630 + 700
= 780 μm
= 1330 μm 2
S = L + 2B
= 780 + 2(15)(11)
= 780 + 330
2
The cell that measures 35 μm by 7 μm by 10 μm will absorb at a greater rate.
15. Find the height of a right cylinder with surface area
160π ft 2 and radius 5 ft.
S = 2πrh + 2πr 2
160π = 2π(5)h + 2π(5)
2
= 1110 μm
ge07se_c10l04003a
AB
16. Find the height of a right rectangular prism with
surface area 286 m 2, length 10 m, and width 8 m.
S = Ph + 2B
286 = 36h + 2(10)(8)
2
286 = 36h + 160
160π = 10πh + 50π
126 = 36h
110π = 10πh
110π
10πh
=
10π
10π
11 = h
_ _
3.5 = h
h = 3.5 m
© Houghton Mifflin Harcourt Publishing Company
h = 11 ft
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
Front/Back Rectangles + Left/Right Rectangles +
Top Front/Back Triangles
(_
12 ft
)
1
S = 2(18 ⋅ 12) + 2(12 ⋅ 12) + 2
⋅ 18 ⋅ 6 = 432 + 288 + 108 = 828 ft 2
2
1 gal
2
828 ft ⋅
≈ 3.3 gal
250 ft 2
Since you can’t get half a gallon, 4 total gallons will be needed.
_
18 ft
4 ⋅ $25 = 100
4 gallons; $100
6
18 ft
12 ft
18
18 ft
12 ft
Module 17
GE_MTXESE353893_U7M17L2 1033
1033
Lesson 17.2
1033
18 ft
Lesson 2
20/01/15 5:53 PM
18. Match the Surface Area with the appropriate coin in the table.
Coin
Diameter (mm)
Thickness (mm)
Surface Area (mm 2)
Penny
19.05
1.55
C
Nickel
21.21
1.95
A
Dime
17.91
1.35
B
Quarter
24.26
1.75
D
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Have students brainstorm how they would find the
surface area of a prism whose dimensions have all
been doubled. Does the surface area double? no If
not, what is the relationship? The area is 4 times as
great. Have students also consider how the surface
area changes if only the height of the prism changes.
Ask students to use examples to justify
their reasoning.
A. 836.58
B. 579.82
C. 662.81
D. 1057.86
Penny L = 2πrh = 2π(9.525)(1.55) = 29.5275π mm 2
S = L + 2πr 2 = 29.5275π + 2π9.525 2 ≈ 662.81 mm 2
Nickel L = 2πrh = 2π(10.605)(1.95) = 41.3595π mm 2
S = L + 2πr 2 = 41.3595π + 2π10.605 2 ≈ 836.58 mm 2
Dime L = 2πrh = 2π(8.955)(1.35) = 24.1785π mm 2
S = L + 2πr 2 = 24.1785π + 2π8.955 2 ≈ 579.82 mm 2
Quarter L = 2πrh = 2π(12.13)(1.75) = 42.455π mm 2
S = L + 2πr 2 = 42.455π + 2π12.13 2 ≈ 1057.86 mm 2
19. Algebra The lateral area of a right rectangular prism is 144 cm 2. Its length is three
times its width, and its height is twice its width. Find its surface area.
w = 3 cm, ℓ = 9 cm, h = 6 cm
ℓ = 3w, h = 2w
S = L + 2B
L = Ph
144 = 2(w + ℓ)h
© Houghton Mifflin Harcourt Publishing Company
= 144 + 2(9)(3)
144 = 2(w + 3w)2w
= 144 + 54
= 198 cm 2
144 = 16w 2
3=w
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.
S = 2πrh + 2πr 2
L = 2πrh
= 2π(8)(3)
= 48π cm
S = L + 2πr 2
= 48π + 2π(8)
= 48π + 128π
= 176π cm 2
Module 17
GE_MTXESE353893_U7M17L2 1034
176π = 2π(4)h + 2π(4)
2
176π = 8πh + 32π
2
2
144π = 8πh
144π ___
____
= 8πh
8π
8π
18 = h
1034
Lesson 2
20/01/15 5:53 PM
Surface Area of Prisms and Cylinders 1034
JOURNAL
H.O.T. Focus on Higher Order Thinking
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
f
t
Have students illustrate and describe how to use
formula S = L + 2B to find the surface area of a
prism and of a right cylinder. Ask them to include all
of the steps as well as the substitutions they will use
in the formula.
10
10
ft
ft
10 ft
10 ft
Surface Area of Triangular Prism (minus bottom side)
Surface Area of Half Cylinder
_1L = _1(2πrh)
2
2
1
=_
(2π(5)(10))
L = Ph
= (30)10
2
= 300 cm 2
= 50π ft 2
_1S = _1L + _12πr 2
2
2
2
1(
= 50π + _
2π5 2)
a2 + b2 = c2
2
5 2 + bge07sec10l04004aa
= 10 2
2
2 pass
25 + b1st
= 100
= 50π + 25π
ge07sec10l04005a
4/23/5
cmurphy
1st pass
= 75π
4/12/5
≈ 235.6 ft 2
cmurphy
b 2 = 75
b = √―
75
S = Ph + 2B - Square
―)
―
= 300 + 5 √75 - 100
(
© Houghton Mifflin Harcourt Publishing Company
1 ( )( √ )
= 300 + 2 _
5
75 - 10 ⋅ 10
2
≈ 243.3 ft 2
The triangular-prism-shaped frame will take more
plastic; In lateral surface area, the area of the bases
are not used. In this case, it is not the area of the
bases that need to be removed.
22. Communicate Mathematical Ideas Explain how to use the net of a
three-dimensional figure to find its surface area.
Find the area of each part of the net, then add the areas.
23. Draw Conclusions Explain how the edge lengths of a rectangular prism
can be changed so that the surface area is multiplied by 9.
Triple all the edge lengths.
Module 17
GE_MTXESE353893_U7M17L2 1035
1035
Lesson 17.2
1035
Lesson 2
22/02/14 5:02 AM
Lesson Performance Task
AVOID COMMON ERRORS
To find the length of a diagonal of one side of the
cube, students must use the Pythagorean Theorem to
find h, the hypotenuse of a right triangle with 2-inch
sides, and then must simplify the resulting square
root. Here are the steps:
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.
h2 = 22 + 22
=4+4
1. The cube has 6 faces each with an area of 2 × 2 = 4 in 2.
Total surface area of the cube: 6 × 4 in 2 = 24 in 2
=8
―
――
= √2 · 2
― 2
= √2 √―
= 2 √―
2
2. The top and bottom of the cylinder are circles, each
with a diameter equal to a diagonal of one side of the
cube, or 2 √2 inches.
―
h = √8
―
2
The radius of the top and bottom is half the diameter,
or √2 inches.
―
2
Area of cylinder top = πr 2 = 3.14( √2 ) = 6.28. Total
area of top and bottom: 2 × 6.28 = 12.56 in 2
2
―
―
―
Total surface area of cylindrical box: (12.56 + 12.56 √2 ) in
Lateral area of cylinder: 2πrh = 2(3.14) √2 (2) = 12.56 √2 in 2
2
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
© Houghton Mifflin Harcourt Publishing Company
Describe how you could find the volume of an empty
cylindrical number-cube container. Then find that
volume. Use 3.14 for π. Subtract the volume of a
number cube from the volume of a cylindrical
container; about 4.56 cubic inches.
V (cylinder) - V (cube) = πr 2h - s 3
―
3
≈ 3.14( √2 ) (2) - (2)
2
= 3.14(2)(2) - 8
= 4.56 in 3
Module 17
1036
Lesson 2
EXTENSION ACTIVITY
GE_MTXESE353893_U7M17L2 1036
A packaging engineer is designing a rectangular-prism-shaped container with a
surface area of 64 square inches. Find the possible dimensions for at least three
containers that have surface areas of 64 square inches.
20/01/15 5:53 PM
Possible dimensions: 4 × 4 × 4; 8 × 2 × 1.6; 6 × 2 × 2.5
Find the volumes of your containers. Then propose a hypothesis about the shape
of a rectangular prism with the greatest volume for a given surface area.
Sample answer: The rectangular prism with the greatest volume for a given
surface area is a cube.
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Surface Area of Prisms and Cylinders 1036