Surface Areas of Prisms and Cylinders

11-2
11-2
Surface Areas of Prisms
and Cylinders
1. Plan
Objectives
1
2
To find the surface area of a
prism
To find the surface area of a
cylinder
Examples
1
2
3
4
Finding Surface Area of
a Prism
Using Formulas to Find
Surface Area
Finding Surface Area of
a Cylinder
Real-World Connection
What You’ll Learn
Check Skills You’ll Need
• To find the surface area of a
Find the area of each net.
prism
• To find the surface area of a
GO for Help
1. 96 cm2
40π cm2
2.
cylinder
4 cm
4 cm
. . . And Why
3.
Lessons 1-9 and 10-3
36"3 m2
8 cm
4 cm
To find the area covered by a
drum on a roller used in road
construction, as in Example 4
6m
4 cm
New Vocabulary • prism • bases, lateral faces, altitude, height, lateral area,
surface area (of a prism) • right prism • oblique prism
• cylinder • bases, altitude, height, lateral area, surface
area (of a cylinder) • right cylinder • oblique cylinder
Math Background
This lesson uses the area formulas
from Chapter 10 and nets to
develop formulas for the lateral
and surface areas of prisms and
cylinders. Nets especially simplify
finding these areas for nonright
prisms, such as parallelepipeds.
The key idea is that the lateral
faces of any prism are
parallelograms.
1
Finding Surface Area of a Prism
A prism is a polyhedron with
exactly two congruent, parallel
faces, called bases. Other faces
are lateral faces. You name a
prism by the shape of its bases. Bases
Lateral
edges
Bases
Lateral
faces
More Math Background: p. 596C
An altitude of a prism is a
perpendicular segment that
joins the planes of the bases.
The height h of the prism is
the length of an altitude.
Lesson Planning and
Resources
A prism may either be right or oblique.
See p. 596E for a list of the
resources that support this lesson.
Real-World
Pentagonal prism
Triangular prism
h
Connection
h
A triangular prism
breaks white light
into rainbow colors.
h
right prisms
oblique prism
PowerPoint
In a right prism the lateral faces are rectangles and a lateral edge is an altitude.
In this book you may assume that a prism is a right prism unless stated or
pictured otherwise.
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
The lateral area of a prism is the sum of the areas of the lateral faces. The
surface area is the sum of the lateral area and the area of the two bases.
Finding Area
Lesson 1-9: Examples 4–6
Extra Skills, Word Problems, Proof
Practice, Ch. 1
Areas of Regular Polygons
Lesson 10-3: Example 2
Extra Skills, Word Problems, Proof
Practice, Ch. 10
608
Chapter 11 Surface Area and Volume
Special Needs
Below Level
L1
Draw examples of prisms on the board and have
students identify the bases, lateral faces, and
altitudes. Point out that in a rectangular prism, any
pair of opposite faces could be considered the bases.
608
learning style: visual
L2
Making nets of rectangular prisms may help students
understand and remember Theorem 11-1.
learning style: tactile
1
2. Teach
Finding Surface Area of a Prism
EXAMPLE
Use a net to find the surface area of the prism at the left.
4
3
Surface Area = Lateral Area + area of bases
= sum of areas of lateral faces + area of bases
5
= (5 ? 4 + 5 ? 3 + 5 ? 4 + 5 ? 3) + 2(3)(4)
5 cm
4
3
Some students may think that a
right prism must contain a right
angle in its base. Point out that
the right angle is formed by the
lateral faces meeting the base.
The surface area of the prism is 94 cm2.
Quick Check
1 Use a net to find the surface area
of the triangular prism. See left.
5 cm
5 cm
1. 216 cm2
6 cm
2
12 cm
You can find formulas for lateral and surface areas by looking at a net for a prism.
5 cm
Perimeter of base
abcd
a
b
c
d
12 cm
c
d Base
a
b
c
d
a
h
b
h
c
Guided Instruction
Error Prevention!
4
= 94
4 cm
3
3
= 70 + 24
3 cm
4
EXAMPLE
Alternative Method
Have students calculate the sum
of the areas of the lateral faces
to help them understand why the
formula L.A. = ph makes sense.
d
PowerPoint
h
Additional Examples
6 cm
Perimeter
Lateral Area ph
Height
Area of
d Base Lateral
Area
a base
c
Surface Area L.A. 2B
1 Use a net to find the surface
area of the cube.
You can use the formulas with any right prism.
2
EXAMPLE
Using Formulas to Find Surface Area
11 in.
Multiple Choice What is the surface area of the prism?
B
A
1
4
5
A
A
B
B
C
By the Pythagorean Theorem, the hypotenuse of the
triangular base is 5 cm.
E
D
C
84 cm2
726 in.2
96 cm2
2 Find the surface area of a
10-cm high right prism with
triangular bases having 18-cm
edges. Round to the nearest
whole number. 821 cm2
E
D
C
B
78 cm2
E
D
C
B
A
3
C
B
A
2
72 cm2
E
D
C
D
D
E
E
Test-Taking Tip
A question could ask
for either surface area
or lateral area of a
solid. Read the
question carefully.
L.A. = ph
= 12 ? 6
3 cm
4 cm
Use the formula for lateral area.
p ≠ 3 ± 4 ± 5 ≠ 12 cm
6 cm
= 72
The lateral area of the prism is 72 cm 2.
Now use the formula for surface area.
S.A. = L.A. + 2B
= 72 + 2(6) = 84
B ≠ 12 (3 ? 4) ≠ 6 cm2
The surface area of the prism is 84 cm2. Choice C is the answer.
Quick Check
2 Use formulas to find the lateral area and surface area
of the prism.
432 m2; about 619 m2
12 m
6m
Lesson 11-2 Surface Areas of Prisms and Cylinders
Advanced Learners
609
English Language Learners ELL
L4
After Example 2, ask students to write a formula for
the surface area of a rectangular prism with edges of
length <, w, and h. S.A. ≠ 2(h< ± hw ± <w)
learning style: verbal
To help students with the new vocabulary, have them
make and display a poster of a prism and a cylinder,
labeling their parts. Clarify that the base of a prism is
a face but the base of a polygon is a side.
learning style: visual
609
Guided Instruction
Key Concepts
Theorem 11-1
Tactile Learners
Lateral and Surface Areas of a Prism
The lateral area of a right prism is the product of
the perimeter of the base and the height.
Have students tape the sides of a
sheet of paper together (without
overlapping) to form a cylinder.
Ask: What is the lateral area of
the cylinder? the area of the
paper
L.A. = ph
p is the
perimeter
of a base.
h
B is the area of a base.
The surface area of a right prism is the sum of
the lateral area and the areas of the two bases.
S.A. = L.A. + 2B
Connection to Algebra
The formula for the surface area
of a cylinder is sometimes written
as S.A. = 2pr(r + h). Have
students show that this formula
is equivalent to the formula
S.A. = 2prh + 2pr2.
4
EXAMPLE
2
1
Finding Surface Area of a Cylinder
Like a prism, a cylinder has two congruent parallel bases. However, the bases of
a cylinder are circles. An altitude of a cylinder is a perpendicular segment that
joins the planes of the bases. The height h of a cylinder is the length of an altitude.
Diversity
Some students may never
have seen a steamroller. Have
other students explain how
steamrollers work.
h
Bases
PowerPoint
h
h
right cylinders
Additional Examples
oblique cylinder
In this book you may assume that a cylinder is a right cylinder unless
stated or pictured otherwise.
3 The radius of the base of a
cylinder is 6 ft, and its height is
9 ft. Find its surface area in terms
of p. 180π ft2
To find the area of the curved surface of a cylinder, visualize “unrolling” it. The
area of the resulting rectangle is the lateral area of the cylinder. The surface area
of a cylinder is the sum of the lateral area and the areas of the two circular bases.
You can find formulas for these areas by looking at a net for a cylinder.
r
Real-World
Connection
A full turn of the roller inks a
rectangle with area equal to
the roller’s lateral area.
Key Concepts
Lateral
Area
h
h
Surface
Area
2pr
μ
r
r
Area of a base
B pr 2
Theorem 11-2
Lateral and Surface Areas of a Cylinder
The lateral area of a right cylinder is the product of the
circumference of the base and the height of the cylinder.
B is the area
of a base. r
L.A. = 2prh, or L.A. = pdh
The surface area of a right cylinder is the sum of the lateral
area and the areas of the two bases.
S.A. = L.A. + 2B, or S.A. = 2prh + 2pr2
610
610
Chapter 11 Surface Area and Volume
h
3
Finding Surface Area of a Cylinder
EXAMPLE
The radius of the base of a cylinder is 4 in. and its height is 6 in. Find the surface
area of the cylinder in terms of p.
S.A. = L.A. + 2B
= 2prh +
Use the formula for surface area of a cylinder.
2(pr 2)
Substitute the formulas for lateral area and area of a circle.
= 2p(4)(6) + 2p(4 2) Substitute 4 for r and 6 for h.
= 48p + 32p
Simplify.
= 80p
Resources
• Daily Notetaking Guide 11-2
The surface area of the cylinder is 80p in.2.
Quick Check
L3
3 Find the surface area of a cylinder with height 10 cm and radius 10 cm in terms of p.
400π cm2
4
Real-World
EXAMPLE
Connection
The area covered is the lateral area of a cylinder that has a diameter of 4.2 ft and a
height of 3.5 ft.
Use the formula for lateral area of a cylinder.
= p(4.2)(3.5)
Substitute.
=
Use a calculator.
46.181412
In one full turn, the large drum covers about 46 ft2.
Quick Check
4 The small drum has diameter 3 ft. 33 ft2
a. To the nearest square foot, what area does the small drum cover in one turn?
b. Critical Thinking What area does the small drum cover in one turn of the
large drum? same as large drum (about 46 ft2)
EXERCISES
• Daily Notetaking Guide 11-2—
L1
Adapted Instruction
Closure
Machinery The drums of the roller at the left are cylinders of length 3.5 ft. The
diameter of the large drum is 4.2 ft. What area does the large drum cover in one
full turn? Round your answer to the nearest square foot.
L.A. = pdh
4 A company sells cornmeal and
barley in cylindrical containers.
The diameter of the base of the
6-in. high cornmeal container is
4 in. The diameter of the base of
the 4-in. high barley container
is 6 in. Which container has the
greater surface area? barley
container
Explain why the factor 2 appears
in both terms of the formula for
the surface area of a cylinder
S.A. = 2prh + 2pr2. 2πrh is the
lateral area, which is the area of
a rectangle with one side equal
to 2πr, the circumference of the
base of the cylinder. The term
2πr2 is the sum of the areas of
the two circular bases of the
cylinder.
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
Use a net to find the surface area of each prism. 1–3. See margin for drawings.
1.
2.
3.
(page 609)
6 ft
29 cm
6 ft
4 in.
6 ft
216 ft2
6.5 cm
19 cm
1726 cm2
8 in.
4 in.
80 ± 32 "2 in.2 or
about 125.3 in.2
4. a. Classify the prism. right hexagonal prism
b. Find the lateral area of the prism. 240 cm2
c. The bases are regular hexagons. Find
10 cm
the sum of their areas. 48 "3 cm2
4 cm
d. Find the surface area of the prism.
(240 ± 48 "3) cm2
Lesson 11-2 Surface Areas of Prisms and Cylinders
6.5 cm
1.
2.
3.
6 ft
6 ft
4 in.
4 in.
29 cm
6.5 cm
19 cm
6 ft
611
8 in.
4 "2 in.
4 "2 in.
611
3. Practice
Example 2
(page 609)
Use formulas to find the lateral area and surface area of each prism. Show your
answer to the nearest whole number.
880 cm2; 1121 cm2
5.
Assignment Guide
8-15, 18, 22, 25,
27-32
C Challenge
33-37
Example 3
(page 611)
8.
Example 4
(page 611)
Exercise 22 Point out that the
B
coordinates are listed in the
order (x, y, z).
L4
L2
Reteaching
L1
Adapted Practice
Class
L3
Date
Practice 11-2
Chords and Arcs
Apply Your Skills
18. A cylinder and a
prism both have two
O n bases and lateral
faces that are
rectangular. The
bases of a cylinder
are circles and the
bases of a prism are
polygons.
21d. 438 units2;
1752 units2; 4 : 1
0
Find the radius and m AB .
2.
A
3.
2 cm
C
B
A
5 cm
C
6 in.
B
GO
B
6 in.
Find the value of x to the nearest tenth.
4.
5.
6.
x
C
C
x
5
2
C
7
3
8.
9.
C
x
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-1102
x
x
3
7.
12
7
3
8
x
14.
6 12 in.
612
© Pearson Education, Inc. All rights reserved.
Q
S
R
A
U
J
C
21. Consider a box with dimensions 3, 4, and 5.
a. Find its surface area. 94 units2
b. Double each dimension and then find the new surface area. 376 units2
c. Find the ratio of the new surface area to the original surface area. 4 : 1
d. Repeat parts (a)–(c) for a box with dimensions 6, 9, and 11. See left.
e. Make a Conjecture How does doubling the dimensions of a rectangular
prism affect the surface area? The surface area becomes 4 times as large.
Chapter 11 Surface Area and Volume
20. Answers may vary.
Sample:
L
A
12. Prove Theorem 11-5, part (2).
Given: 䉺O, OE # AB, OF # CD, AB = CD
Prove: OE = OF
4 cm
3 cm
E
O
B
C
F
0
0
0
13. Given: 䉺O with mAB = mBC = mCA
Prove: m⬔ABC = m⬔BCA = m⬔CAB
D
A
14 cm
O
B
C
612
1407 cm2
19. A hexagonal pencil is a hexagonal prism. A base
edge of the pencil has length 4 mm. The pencil
(without eraser) has height 170 mm. How much
surface area of a hexagonal pencil gets painted?
4080 mm2
20. Open-Ended Draw a net for a rectangular
prism with a surface area of 220 cm 2. See margin.
䉺A 䉺J, BC KL
B
K
Write a two-column proof, a paragraph proof, or a flow proof.
20 cm
17. Estimation Estimate the surface area of a cube with edges 4.95 cm long.
150 cm2
18. Writing Explain how a cylinder and a prism are alike and how they
are different. See left.
T
P
8 cm
16. A triangular prism has base edges 4 cm, 5 cm, and 6 cm long. Its lateral area
is 300 cm2. What is the height of the prism? 20 cm
20
11.
䉺Q 䉺T, PR SU
9m
226 m2
C
86ⴗ
List what you can conclude from each diagram.
10.
15.
6m
6
C
x
120ⴗ
11 in.
101.5π in.2
16.5π cm2
107 in.2
Exercise 26 Point out that the
C
4 cm
4 in.
Orange
Juice
Connection to
Coordinate Geometry
Enrichment
7 in.
12. Packaging A cylindrical carton of oatmeal with radius 3.5 in. is 9 in. tall. If all
surfaces except the top are made of cardboard, how much cardboard is used to
make the oatmeal carton? Round your answer to the nearest square inch.
236.4 in.2
Find the surface area of each cylinder to the nearest whole number.
13.
surface area available for the
label is in fact the lateral area of
the can.
L3
Regular
octagon
11. A standard drinking straw is 19.5 cm long and has a diameter of 0.6 cm. How
many square centimeters of plastic are used in one straw? Round your answer
to the nearest tenth. 36.8 cm2
Exercise 2 After students finish
the exercise, ask: What formula
gives the surface area of a cube
with sides of length s? S.A. ≠ 6s2
24
ft
5 ft
10.
3 cm
40π cm2
Connection to Algebra
1. A
9.
2 cm
8 cm
Homework Quick Check
GPS Guided Problem Solving
5 cm
Find the surface area of each cylinder in terms of π.
38-41
42-46
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 12, 18, 24, 26.
22 cm
5 in.
96 in.2; 108 in.2
10 ft
120 ft 2; 220 ft 2
2 A B
Name
8 in.
5 ft
24, 26
Practice
3 in. 7.
4 in.
1 A B 1-7, 16, 17, 19-21, 23,
Test Prep
Mixed Review
6.
4 ft
4 cm
22. Multiple Choice A cylindrical can of cocoa has the
dimensions shown at the right. What is the
approximate surface area available for the label? A
110 in.2
148 in.2
2
179 in.
219 in.2
23. Pest Control A flour moth trap has the shape of
a triangular prism that is open on both ends. An
environmentally safe chemical draws the moth
inside the prism, which is lined with an adhesive.
Find the surface area of the trap. 47.5 in.2
1 in.
7 in.
C O C O A 5 in.
4. Assess & Reteach
PowerPoint
Lesson Quiz
2 in.
4 in.
Use the prism below for Exercises
1 and 2.
5 in.
3.5 in.
24. Packaging A typical box for a videocassette tape is open on one side as
GPS pictured at the left. How many square inches of cardboard are in a typical box
for a videocassette tape? about 75.5 in.2
712 in.
4 in.
Exercise 24
27. cylinder of radius 4
and height 2;
48π units2
25. Suppose that a cylinder has a radius of r units, and that the height of the
cylinder is also r units. The lateral area of the cylinder is 98p square units.
x 2 a. Algebra Find the value of r. 7 units
b. Find the surface area of the cylinder. 196π units2
z
26. a. Geometry in 3 Dimensions Find the three
D
coordinates of each vertex A, B, C, and D
3
of the rectangular prism. A(3, 0, 0); B(3, 5, 0);
1
C(0, 5, 0); D(0, 5, 4)
b. Find AB. 5
C
1
c. Find BC. 3
y
O 2
4
d. Find CD. 4
3
B
e. Find the surface area of the prism. 94 units2 x A
28. cylinder of radius 2
and height 4;
24π units2
Visualization The plane region is revolved
completely about the given line to sweep out
a solid of revolution. Describe the solid and
find its surface area in terms of π. 27–30. See left.
29. cylinder of radius 2
and height 4;
24π units2
27. the y-axis
28. the x-axis
29. the line y = 2
30. the line x = 4
30. cylinder of radius 4
and height 2;
48π units2
GO for Help
For a guide to solving
Exercise 32, see p. 615.
31b. Surface area is
more than doubled.
C
Challenge
31c. S.A. ≠
± 2πrh;
if r doubles:
S.A. ≠ 2(4πr 2 ±
2πrh). Since r is
squared, surface
area is more than
doubled.
2πr 2
3
2
1
O
m
7c
34.
4m
x
1 2 3 4
8 cm
(148 ± 66.5π) cm2
lesson quiz, PHSchool.com, Web Code: aua-1102
6m
3m
(84 ± 20π) m2
1. Use a net to find the surface
area.
8 ft
10 ft
8 ft
7 ft
6 ft
7 ft
6 ft
8 ft
7.5 in.
35.
8 in.
4. The radius of the base
of a cylinder is 16 in., and its
height is 4 in. Find its surface
area in terms of p. 640π in.2
Alternative Assessment
Have partners design and label
two different prisms, each having
a surface area of 200 cm2.
10 in.
(220 – 8π) in.2
Lesson 11-2 Surface Areas of Prisms and Cylinders
3. The height of a prism is
5 cm. Its rectangular bases
have 3-cm and 9-cm sides. Find
its surface area. 174 cm2
5. A contractor paints all but the
bases of a 28-ft high cylindrical
water tank. The diameter of
the base is 22 ft. How many
square feet are painted?
Round to the nearest hundred.
1900 ft2
6 in.
3 in.
4 cm
7 ft
2. Use a formula to find
the surface area.
S.A. ≠ L.A. ± 2B ≠
168 ± 48 ≠ 216; 216 ft2
Judging by appearances, what is the surface area of each solid?
33.
6 ft
S.A. ≠ 216 ft2
y
31. a. Critical Thinking Suppose you double the radius of a right cylinder. How
does that affect the lateral area? Lateral area is doubled.
b. How does that affect the surface area? b-c. See left below.
c. Use the formula for surface area of a right cylinder to explain why the
surface area in part (b) was not doubled.
32. a. Packaging Some cylinders have wrappers
with a spiral seam. Peeled off, the wrapper
has the shape of a parallelogram. The
wrapper for a biscuit container has base
7.5 in. and height 6 in. Find the radius and
height of the container. r < 1.2 in.; h ≠ 6 in.
b. Find the surface area of the container. about 54 in.2
8 ft
613
613
Test Prep
x 2 36. Algebra The sum of the height and radius of a cylinder is 9 m. The surface area
of the cylinder is 54p m 2. Find the height and the radius. h ≠ 6 m; r ≠ 3 m
Resources
37. Each edge of the large cube at the right is
12 inches long. The cube is painted on the
outside, and then cut into 27 smaller cubes.
Answer these questions about the 27 cubes.
a. How many are painted on 4, 3, 2, 1, and 0 faces?
b. What is the total surface area that is unpainted?
b. 1728 in.2
a. 0, 8, 12, 6, 1
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 657
• Test-Taking Strategies, p. 652
• Test-Taking Strategies with
Transparencies
Test Prep
Multiple Choice
41. [2] a. Lateral area is
the perimeter
times height.
Since the lateral
area is 48 in.2 and
the perimeter of
the k is 24 in.,
h ≠ 2 in.
b. S.A. ≠ 96 in.2
[1] correct explanation
correct surface area
Short Response
38. What is the surface area of the figure
to the nearest tenth? C
A. 335.7 m2
B. 411.6 m2
2
C. 671.5 m
D. 721.2 m2
7.4 m
6.8 m
20.1 m
39. If the radius and height of a cylinder are
both doubled, then the surface area is 9. J
F. the same
G. doubled
H. tripled
J. quadrupled
40. A cylinder of radius r sits snugly inside a cube. Which expression represents
the difference of their lateral areas? D
A. 2r2(8 - p)
B. 2r(p - 2)
C. 2r(4 - p)
D. 4r2(4 - p)
41. The sides of a base of a right triangular prism are 6 in., 8 in., and 10 in. The
lateral area of the prism is 48 in.2. a–b. See left.
a. Find the height of the prism. Explain your reasoning.
b. What is the surface area of the prism?
Mixed Review
Lesson 11-1
GO for
Help
Sketch each space figure and then draw a net for it. Label the net with
its dimensions.
42. a rectangular box with height 5 cm and a base 3 cm by 4 cm
42–43. See back of book.
43. a cube with 2-in. sides
Lesson 10-7
Find the area of each part of the circle to the nearest tenth.
44. sector QOP 37.7 cm2
Lesson 6-7
0
45. the segment of the circle bounded by QP and QP
22.1 cm2
46. In the kite at the right AB = AD and CB = CD.
Points P, Q, R, and S are midpoints.
a. Determine the coordinates of the
R(c, b), S(c, –b), C(2c, 0) R
midpoints.
b. RQ = ■; SP = ■; P(a, –b), Q(a, b)
PQ = ■; SR = ■ a – c; a – c; 2b; 2b
c. Use your answers to part (b) to explain
why PQRS must be a parallelogram.
Opposite sides are O.
614
614
Chapter 11 Surface Area and Volume
S
P
Q
120
cm
O 6
B (0, 2b)
Q
O
P
D (0, -2b)
A (2a, 0)