GETE1102.pdf

11-2
1. Plan
Objectives
1 To find the surface area of a
prism
2 To find the surface area of a
cylinder
Examples
1 Finding Surface Area of
a Prism
2 Using Formulas to Find
Surface Area
3 Finding Surface Area of
a Cylinder
4 Real-World Connection
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.
More Math Background: p. 596C
Lesson Planning and
Resources
See p. 596E for a list of the
resources that support this lesson.
PowerPoint
608
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
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
11-2
What You’ll Learn
• To find the surface area of a
prism
• To find the surface area of a
cylinder
. . . And Why
To find the area covered by a
drum on a roller used in road
construction, as in Example 4
1 Finding Surface Area of a Prism
Real-World Connection
A triangular prism
breaks white light
into rainbow colors.
608 Chapter 11 Surface Area and Volume
Special Needs 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.
Surface Areas of Prisms
and Cylinders
Check Skills You’ll Need GO for Help
Find the area of each net.
1. 96 cm 2.
4 cm
3.
2
40 cm2 π 36 m2 "3
4 cm
4 cm
4 � cm
8 cm
Lessons 1-9 and 10-3
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
A prism is a polyhedron with
exactly two congruent, parallel
faces, called bases. Other faces
are lateral faces. You name a
Lateral
edges
prism by the shape of its bases. Bases
Bases
An altitude of a prism is a
Lateral
perpendicular segment that
joins the planes of the bases.
faces
The height h of the prism is
the length of an altitude.
Pentagonal prism Triangular prism
A prism may either be right or oblique.
h
right prisms
oblique prism
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.
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.
h
h
6 m
Below Level L2
Making nets of rectangular prisms may help students
understand and remember Theorem 11-1.
learning style: visual learning style: tactile

5 cm
1. 216 cm 2
1 A B C D E
2 A B C D E
3 A B C D E
4 A B C D E
5 A B C D E
B C D E
6 cm
4 cm
Quick Check
5 cm
Test-Taking Tip
3 cm
A question could ask
for either surface area
or lateral area of a
solid. Read the
question carefully.
12 cm
Quick Check
1
2
2
EXAMPLE
Finding Surface Area of a Prism
Use a net to find the surface area of the prism at the left.
Surface Area = Lateral Area + area of bases
= sum of areas of lateral faces + area of bases
= (5 ? 4 + 5 ? 3 + 5 ? 4 + 5 ? 3) + 2(3)(4)
= 70 + 24
= 94
The surface area of the prism is 94 cm2 .
1 Use a net to find the surface area
of the triangular prism. See left.
You can find formulas for lateral and surface areas by looking at a net for a prism.
Perimeter of base
a � b � c � d d
c
d
c
Base
h
a b c d a b
h
a b c d
h
Perimeter Height
Lateral Area � ph
You can use the formulas with any right prism.
EXAMPLE
Using Formulas to Find Surface Area
Multiple Choice What is the surface area of the prism?
72 cm 2 78 cm 2 84 cm 2 96 cm 2
By the Pythagorean Theorem, the hypotenuse of the
triangular base is 5 cm.
L.A. = ph Use the formula for lateral area.
= 12 ? 6 p ≠ 3 ± 4 ± 5 ≠ 12 cm
= 72
The lateral area of the prism is 72 cm2 .
Now use the formula for surface area.
S.A. = L.A. + 2B
5 cm
6 cm
= 72 + 2(6) = 84 B ≠ (3 ? 4) ≠ 6 cm2 1
2
The surface area of the prism is 84 cm 2 . Choice C is the answer.
Use formulas to find the lateral area and surface area
of the prism.
432 m 2 ; about 619 m 2
Advanced Learners L4
After Example 2, ask students to write a formula for
the surface area of a rectangular prism with edges of
length

Guided Instruction
Tactile Learners
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
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 + 2pr 2 .
4
610
EXAMPLE
Diversity
Some students may never
have seen a steamroller. Have
other students explain how
steamrollers work.
PowerPoint
Additional Examples
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 3
12 Finding Surface Area of a Cylinder
Real-World
Key Concepts Theorem 11-1 Lateral and Surface Areas of a Prism
Connection
A full turn of the roller inks a
rectangle with area equal to
the roller’s lateral area.
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.
Bases
h
right cylinders
oblique cylinder
In this book you may assume that a cylinder is a right cylinder unless
stated or pictured otherwise.
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.
Key Concepts Theorem 11-2 Lateral and Surface Areas of a Cylinder
610 Chapter 11 Surface Area and Volume
h
The lateral area of a right prism is the product of
the perimeter of the base and the height.
L.A. = ph
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
2pr
Lateral
Area
The lateral area of a right cylinder is the product of the
circumference of the base and the height of the cylinder.
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 h
r
h
Surface
Area
h
μ
r
r
h
p is the
perimeter
of a base.
B is the area of a base.
h
Area of a base
B � pr 2
B is the area
of a base. r

Quick Check
Quick Check
Example 1
(page 609)
3
4
EXAMPLE
Finding Surface Area of a Cylinder
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 Use the formula for surface area of a cylinder.
= 2prh + 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
The surface area of the cylinder is 80p in. 2 .
3 Find the surface area of a cylinder with height 10 cm and radius 10 cm in terms of p.
400 cm2 π
EXAMPLE Real-World Connection
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.
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.
L.A. = pdh Use the formula for lateral area of a cylinder.
= p(4.2)(3.5) Substitute.
= 46.181412 Use a calculator.
In one full turn, the large drum covers about 46 ft 2 .
The small drum has diameter 3 ft.
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 4
33 ft
)
2
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
GO for
Help
Use a net to find the surface area of each prism.
1. 2. 3.
4. a. Classify the prism.
b. Find the lateral area of the prism.
c. The bases are regular hexagons. Find
the sum of their areas. 48 cm2 6 ft
29 cm
6 ft
6 ft
4 in.
8 in.
6.5 cm
19 cm
1726 cm
4 in.
"3
d. Find the surface area of the prism.
4 cm
10 cm
2
216 ft2 1–3. See margin for drawings.
right hexagonal prism
80 ± 32 in. 2 or
about 125.3 in. 2
240 cm
"2
2
(240 ± 48 ) cm2 "3
Lesson 11-2 Surface Areas of Prisms and Cylinders 611
1. 6.5 cm
2. 3.
6 ft
6 ft
29 cm
6.5 cm
19 cm
6 ft
4 in.
8 in.
4 in.
4 "2
in.
4 "2 in.
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
Resources
• Daily Notetaking Guide 11-2
L3
• Daily Notetaking Guide 11-2—
Adapted Instruction L1
Closure
Explain why the factor 2 appears
in both terms of the formula for
the surface area of a cylinder
S.A. = 2prh + 2pr 2 . 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
2πr 2 is the sum of the areas of
the two circular bases of the
cylinder.
611

3. Practice
Assignment Guide
1
612
A B 1-7, 16, 17, 19-21, 23,
24, 26
2 A B 8-15, 18, 22, 25,
27-32
C Challenge 33-37
Test Prep 38-41
Mixed Review 42-46
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 12, 18, 24, 26.
Connection to Algebra
Exercise 2 After students finish
the exercise, ask: What formula
gives the surface area of a cube
with sides of length s? S.A. ≠ 6s 2
Exercise 22 Point out that the
surface area available for the
label is in fact the lateral area of
the can.
Connection to
Coordinate Geometry
Exercise 26 Point out that the
coordinates are listed in the
order (x, y, z).
GPS
Enrichment
Guided Problem Solving
Reteaching
Adapted Practice
Practice
© Pearson Education, Inc. All rights reserved.
Name Class Date
Practice 11-2
0
Find the radius and m AB .
1. 2. 3.
A
24 ft
5 ft
C
Find the value of x to the nearest tenth.
7. 8. 9.
C
12
x
120�
B
4. 5. x
6.
C
x
5
8
2
C
3
C
7 7
3
x
3 x
List what you can conclude from each diagram.
10. �Q � �T, PR � SU
11. �A � �J, BC � KL
B
K
Q
T
P
S
A
J
R
U
C
L
Write a two-column proof, a paragraph proof, or a flow proof.
12. Prove Theorem 11-5, part (2).
Given: �O, OE # AB,
OF # CD,AB=
CD
Prove: OE = OF
0 0 0
13. Given: �O with m AB = m BC = mCA
Prove: m�ABC = m�BCA = m�CAB
20
C
6 in.
A B
6 in.
x
C
86�
2 cm
C
x
6
C
A
B
A
E
O
B
C
F
D
C
A
O
B
5 cm
L4
L1
L3
L2
L3
Chords and Arcs
B
Example 2
(page 609)
Example 3
(page 611)
Example 4
(page 611)
Apply Your Skills
18. A cylinder and a
prism both have two
Onbases 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
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-1102
612 Chapter 11 Surface Area and Volume
20. Answers may vary.
Sample:
Use formulas to find the lateral area and surface area of each prism. Show your
answer to the nearest whole number.
880 cm
5. 4 ft 6. 3 in. 7.
2 ; 1121 cm2 4 in.
8 in.
10 ft
5 ft
5 in.
120 ft2 ; 220 ft2 96 in. 2 ; 108 in. 2
Find the surface area of each cylinder in terms of π.
8. 2 cm
9. 3 cm
10.
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 cm 2
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.
Find the surface area of each cylinder to the nearest whole number.
13. 4 in.
14. 15.
8 cm
2
Orange
Juice
8 cm
40π cm 2
107 in. 2
1
6
2
in.
4 cm
6 m 9 m
226 m 2
16.5π cm 2
22 cm
5 cm Regular
octagon
20 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
17. Estimation Estimate the surface area of a cube with edges 4.95 cm long.
18. Writing Explain how a cylinder and a prism are alike and how they
are different. See left.
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?
20. Open-Ended Draw a net for a rectangular
prism with a surface area of 220 cm2 . See margin.
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 150 cm
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.
e. Make a Conjecture How does doubling the dimensions of a rectangular
prism affect the surface area?
2
4080 mm2 See left.
The surface area becomes 4 times as large.
4 cm
3 cm
4 cm
14 cm
7 in.
11 in.
101.5π in. 2
1407 cm 2

1 in.
7 1 2 in.
4 in.
Exercise 24
27. cylinder of radius 4
and height 2;
48π units 2
28. cylinder of radius 2
and height 4;
24π units 2
29. cylinder of radius 2
and height 4;
24π units 2
30. cylinder of radius 4
and height 2;
48π units 2
GO
for Help
For a guide to solving
Exercise 32, see p. 615.
31b. Surface area is
more than doubled.
C
Challenge
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
179 in. 2 219 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
24. Packaging A typical box for a videocassette tape is open on one side as
pictured at the left. How many square inches of cardboard are in a typical box
for a videocassette tape? about 75.5 in. 2
GPS
25. Suppose that a cylinder has a radius of r units, and that the height of the
x2 x 2
cylinder is also r units. The lateral area of the cylinder is 98p square units.
a. Algebra Find the value of r. 7 units
b. Find the surface area of the cylinder. 196π units 2
26. a. Geometry in 3 Dimensions Find the three
coordinates of each vertex A, B, C, and D
of the rectangular prism.
b. Find AB. 5
c. Find BC. 3
d. Find CD. 4
e. Find the surface area of the prism. 94 units2 A(3, 0, 0); B(3, 5, 0);
C(0, 5, 0); D(0, 5, 4)
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.
27. the y-axis 28. the x-axis
29. the line y = 2 30. the line x = 4
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
6 in.
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.
7.5 in.
b. Find the surface area of the container. about 54 in. 2
Judging by appearances, what is the surface area of each solid?
31c. S.A. ≠ 2πr 33. 34. 4 m
35.
8 in.
4 cm
8 cm
3 in.
6 m 3 m
10 in.
Lesson 11-2 Surface Areas of Prisms and Cylinders 613
2 ± 2πrh;
if r doubles:
S.A. ≠ 2(4πr2 ±
2πrh). Since r is
squared, surface
area is more than
doubled.
(148 ± 66.5π) cm2 (84 ± 20π) m2 (220 – 8π) in. 2
lesson quiz, PHSchool.com, Web Code: aua-1102
7 cm
x
2 in.
z
1
1 O
3
A
C O C O A
3
3
2
1
O
4 in.
3.5 in.
y
1
5 in.
C
2 4 y
B
2 3 4
5 in.
7 in.
D
x
4. Assess & Reteach
PowerPoint
Lesson Quiz
Use the prism below for Exercises
1 and 2.
1. Use a net to find the surface
area.
8 ft
8 ft
10 ft
6 ft
7 ft 7 ft
6 ft
8 ft
8 ft
6 ft
S.A. ≠ 216 ft 2
7 ft
2. Use a formula to find
the surface area.
S.A. ≠ L.A. ± 2B ≠
168 ± 48 ≠ 216; 216 ft 2
3. The height of a prism is
5 cm. Its rectangular bases
have 3-cm and 9-cm sides. Find
its surface area. 174 cm 2
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
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 ft 2
Alternative Assessment
Have partners design and label
two different prisms, each having
a surface area of 200 cm 2 .
613

Test Prep
Resources
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
614
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
Mixed Review
GO for
Help
Lesson 11-1
Lesson 10-7
Lesson 6-7
x2 x 2
614 Chapter 11 Surface Area and Volume
36. Algebra The sum of the height and radius of a cylinder is 9 m. The surface area
of the cylinder is 54p m2 . Find the height and the radius. h ≠ 6 m; r ≠ 3 m
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?
a. 0, 8, 12, 6, 1 b. 1728 in. 2
38. What is the surface area of the figure
to the nearest tenth? C
A. 335.7 m 2 B. 411.6 m 2
C. 671.5 m 2 D. 721.2 m 2
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. 2r 2 (8 - p) B. 2r(p - 2) C. 2r(4 - p) D. 4r 2 (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?
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
Find the area of each part of the circle to the nearest tenth.
44. sector QOP 37.7 cm2 0
45. the segment of the circle bounded by QP and QP
22.1 cm
46. In the kite at the right AB = AD and CB = CD.
P 120�
O
Q
Points P, Q, R, and S are midpoints.
a. Determine the coordinates of the
midpoints.
b. RQ = ■; SP = ■;
C(2c, 0)
R
B (0, 2b)
Q
O A (2a, 0)
PQ = ■; SR = ■ a – c; a – c; 2b; 2b
c. Use your answers to part (b) to explain
why PQRS must be a parallelogram.
S
P
D (0, -2b)
2
R(c, b), S(c, –b),
P(a, –b), Q(a, b)
Opposite sides are O.
6 cm
7.4 m
6.8 m