GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
2. The diameter is 4, so the
3 APPLY
1. Describe the boundaries of a sector of a circle.
2 radii of a › and their intercepted arc
2
1
2. In Example 5 on page 693, explain why the expression π • ᎏᎏ • 4
2
represents the area of the circle cut from the wood. See margin.
冉 冊
ASSIGNMENT GUIDE
BASIC
Day 1: pp. 695–698 Exs. 10–28
even, 30–32, 35–37, 40,
41, 43–44, 46–60 even
In Exercises 3–8, find the area of the shaded region.
3.
4.
radius is half that, or
9 in.
C
1
}} • 4. Then the area is
2
2
1
πr 2 = π }} • 4 .
2
冉 冊
6.
A
5.
A
81π ≈ 254.47 in.2
7.
A
C
A
110ⴗ
C
8.
ADVANCED
36π ≈ 113.10 ft2
10 m
3 in.
70⬚
C
Day 1: pp. 695–698 Exs. 10–28
even, 30–37, 40, 41, 43–45,
46–60 even
60ⴗ
C
BLOCK SCHEDULE WITH 11.6
B
175π
}} ≈ 61.09 m 2
9
B
Day 1: pp. 695–698 Exs. 10–28
even, 30–37, 40, 41, 43–44,
46–60 even
C 12 ft
14.44π ≈ 45.36 cm2
6 ft
AVERAGE
3.8 cm
pp. 695–698 Exs. 10–28 even,
30–37, 40, 41, 43–44, 46–60 even
15π
}} ≈ 23.56 in.2
2
11π ≈ 34.56 ft2
9.
PIECES OF PIZZA Suppose the pizza shown is
divided into 8 equal pieces. The diameter of the pizza
is 16 inches. What is the area of one piece of pizza?
EXERCISE LEVELS
Level A: Easier
10–13
Level B: More Difficult
14–44
Level C: Most Difficult
42, 45
8π ≈ 25.13 in.2
PRACTICE AND APPLICATIONS
STUDENT HELP
FINDING AREA In Exercises 10–18, find the area of the shaded region.
Extra Practice
to help you master
skills is on p. 824.
10.
11.
A
12.
B
31 ft
C
C
C
8m
0.4 cm
A
A
961π ≈ 3019.07 ft2
0.16π ≈ 0.50 cm2
13.
14.
16π ≈ 50.27 m 2
15.
A
A
20 in.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 10–13,
19, 20
Example 2: Exs. 14–18, 21,
22, 29
Example 3: Exs. 14–18, 21,
22, 29
Example 4: Exs. 23–28,
35–37
Example 5: Exs. 23–28,
35–37
Example 6: Exs. 38–40
B
2
100π ≈ 314.16 in.
16.
C
60ⴗ
C
17.
A
1
32
11 ft
121π
}} ≈ 63.36 ft2
6
10 cm
C
293ⴗ
4.6 m
B
B
COMMON ERROR
EXERCISES 10–18 Students
may confuse the formulas for area
of a circle or sector with those
for circumference or arc length.
Tell them that area is measured in
square units, so the radius is
squared in the area formulas.
B
C
E
529π
}} ≈ 22.16 m 2
75
in.
49π
}} ≈ 8.55 in.2
18. 18
A
125⬚
C
1465π
}} ≈ 255.69 cm2
18
80ⴗ
C
8 in.
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 14, 20, 24, 32, 36, 40. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 11
Resource Book, p. 85)
•
Transparency (p. 84)
D
400π
}} ≈ 139.63 in.2
9
19. USING AREA What is the area of a circle with diameter 20 feet?
100π ≈ 314.16 ft2
20. USING AREA What is the radius of a circle with area 50 square meters?
2π
5兹苶
}} ≈ 3.99 m
π
11.5 Areas of Circles and Sectors
695
695
MATHEMATICAL REASONING
EXERCISES 31 AND 32 Ask
students to write the equation representing the relationship in Ex. 31
in terms of ␲. Remind them that ␲ is
a constant so that this is a linear
equation. Then have them write
the equation for the relationship in
Ex. 32. Students should see that this
is another linear function.
ENGLISH LEARNERS
EXERCISE 34 Note that sentences with several consecutive
prepositional phrases can present
problems for some English learners.
You might paraphrase the first
sentence in Exercise 34 so that
students understand the relationship between the boat and the
beam of light.
30.
USING AREA Find the indicated measure. The area given next to the
diagram refers to the shaded region only.
21. Find the radius of ›C. 13.00 in.
C
22. Find the diameter of ›G.
A
40ⴗ Area ⴝ 59 in.2
72ⴗ
G
B
Area ⴝ 277 m2
FINDING AREA Find the area of the shaded region.
25. 16π º 80 cos 36° sin 36°
≈ 12.22 ft2
23.
24.
25.
19 cm
6m
4 ft
180ⴗ
24 m
1083π
}} ≈ 1701.17 cm2
2
540π ≈ 1696.46 m 2
26.
28.
1 ft
2 cm
18 in.
5π ≈ 15.71 ft2
30
x
See margin.
27.
y
2
21.00 m
31. Yes; it appears that the
points lie along a line.
You can also write a
π
linear equation, y = }}x.
40
32. Yes; no; the areas would
change because the
radius of the › changes;
the relationship between
x and y would be linear,
5π
with y = }}x.
72
FOCUS ON
APPLICATIONS
180ⴗ
60ⴗ
18 in.
2π º 兹3苶 ≈ 4.55 cm2
324 º 81π ≈ 69.53 in.2
FINDING A PATTERN In Exercises 29–32, consider an arc of a circle with a
radius of 3 inches.
29. Copy and complete the table. Round your answers to the nearest tenth.
Measure of arc, x
Area of corresponding sector, y
30°
60°
90°
120°
150°
180°
?
?
?
?
?
?
2.4
4.7
7.1
9.4
11.8
14.1
30. xy USING ALGEBRA Graph the data in the table.
See margin.
31. xy USING ALGEBRA Is the relationship between x and y linear? Explain.
See margin.
32.
LOGICAL REASONING If Exercises 29–31 were repeated using a circle
with a 5 inch radius, would the areas in the table change? Would your answer
to Exercise 31 change? Explain your reasoning. See margin.
LIGHTHOUSES The diagram shows a
projected beam of light from a lighthouse.
28 mi
33. What is the area of water that can be
covered by the light from the lighthouse?
692.72 mi2
RE
FE
L
AL I
34. Suppose a boat traveling along a straight
LIGHTHOUSES
use special lenses
that increase the intensity of
the light projected. Some
lenses are 8 feet high and
6 feet in diameter.
696
696
245ⴗ
line is illuminated by the lighthouse for
approximately 28 miles of its route. What
is the closest distance between the lighthouse
and the boat? about 11.31 mi
Chapter 11 Area of Polygons and Circles
lighthouse
18 mi
39. You would need to
know the length of the
diagonals of the kite,
the measure of the
central angle of the
sector, and the radius
of the sector.
USING AREA In Exercises 35–37, find the area of the shaded region in the
circle formed by a chord and its intercepted arc. (Hint: Find the difference
between the areas of a sector and a triangle.)
35.
36.
E
60
6 cm
G
L
37.
A
14 m
P
48 cm
3 3 cm
120
N
C
72 m
STUDENT HELP NOTES
B
F
M
6π º 93 ≈ 3.26 cm2
768π º 5763 ≈ 1415.08 cm2
49π º 98 ≈ 55.94 m2
FOCUS ON
APPLICATIONS
APPLICATION NOTE
EXERCISES 38 AND 39
Additional information about
Viking longships is available at
www.mcdougallittell.com.
VIKING LONGSHIPS Use the information
below for Exercises 38 and 39.
When Vikings constructed longships,
they cut hull-hugging frames from
curved trees. Straight trees provided
angled knees, which were used to brace
the frames.
2
2
angled
knee
Look Back As students look back
to Activity 11.4 on page 690,
remind them to enter formulas in
the first few rows of the spreadsheet so that the rest of the
spreadsheet can be filled in using
the Fill Down command.
2.04 ft or 294.05 in.
38. Find the area of a cross-section of
the frame piece shown in red.
39.
RE
FE
L
AL I
VIKING
LONGSHIPS
INT
The planks in the hull of a
longship were cut in a radial
pattern from a single green
log, providing uniform
resiliency and strength.
40.
NE
ER T
APPLICATION LINK
41.
42.
frame
6 in.
A
B
D
C
R
4 ft
P
S
2 ft T
LOGICAL REASONING Suppose a circle has a radius of 4.5 inches. If
you double the radius of the circle, does the area of the circle double as well?
What happens to the circle’s circumference? Explain.
TECHNOLOGY The area of a regular n-gon
inscribed in a circle with radius 1 unit can
be written as
Look Back
to Activity 11.4 on
p. 690 for help with
spreadsheets.
3 ft
WINDOW DESIGN The window
shown is in the shape of a semicircle
with radius 4 feet. The distance from
S to T is 2 feet, and the measure of
AB is 45°. Find the area of the glass
in the region ABCD. 1.5π ≈ 4.71 ft2
1
180°
A = ᎏᎏ cos ᎏᎏ
2
n
STUDENT HELP
72⬚
៣
www.mcdougallittell.com
41. No; the area of the ›
is quadrupled and the
circumference is
doubled. A = πr 2 and
C = 2πr; π(2r)2 =
4πr 2 = 4A and
2π(2r) = 2(2πr) = 2C.
Writing The angled knee piece
shown in blue has a cross section
whose shape results from subtracting
a sector from a kite. What
measurements would you need to
know to find its area? See margin.
.
180°
2n • sin ᎏᎏ
n
cos
180
n
Use a spreadsheet to make a table. The first
column is for the number of sides n and the
second column is for the area of the n-gon.
Fill in your table up to a 16-gon. What do
you notice as n gets larger and larger?
ADDITIONAL PRACTICE
AND RETEACHING
180
n
sin
For Lesson 11.5:
180n The areas of the polygons approach the value of π, the area of the ›.
11.5 Areas of Circles and Sectors
697
• Practice Levels A, B, and C
(Chapter 11 Resource Book,
p. 75)
• Reteaching with Practice
(Chapter 11 Resource Book,
p. 78)
•
See Lesson 11.5 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
697
Test
Preparation
4 ASSESS
DAILY HOMEWORK QUIZ
★ Challenge
40°
B
EXTRA CHALLENGE
about 8.7 ft 2
3. Find the area of the shaded
region.
B
¡
E
¡
6π
60π
C
¡
9π
108ⴗ
R
S
27π
P
180π
3
q
44. MULTIPLE CHOICE What is the area of the
region shaded in red? D
A
5 ft
43. MULTIPLE CHOICE If ›Q is cut away,
what is the remaining area of ›P? C
A
¡
D
¡
Transparency Available
1. What is the area of a circle
with radius 9 cm? 81␲ cm 2
2. Find the area of the shaded
region.
C
›Q and ›P are tangent. Use the diagram for Exercises 43 and 44.
www.mcdougallittell.com
A
¡
D
¡
B
¡
E
¡
0.3
10.8π
1.8π
C
¡
6π
108π
45. FINDING AREA Find the area between
C
the three congruent tangent circles. The
radius of each circle is 6 centimeters.
(Hint: ¤ABC is equilateral.)
363 º 18π ≈ 5.81 cm2
A
B
MIXED REVIEW
15 in.
SIMPLIFYING RATIOS In Exercises 46–49, simplify the ratio.
(Review 8.1 for 11.6)
8 cats
46. ᎏᎏ
20 cats
6 teachers 3
47. ᎏᎏ }}
32 teachers 16
2
}}
5
52 weeks 4
49. ᎏᎏ }}
143 weeks 11
12 inches 4
48. ᎏᎏ }}
63 inches 21
50. The length of the diagonal of a square is 30. What is the length of each side?
(Review 9.4) 152
1800 – 450␲ in.2
FINDING MEASURES Use the diagram to find
the indicated measure. Round decimals to
the nearest tenth. (Review 9.6)
4. What is the radius of a circle
with area 100␲ square meters?
10 m
EXTRA CHALLENGE NOTE
55. (x + 2)2 + (y + 7)2 = 36
Challenge problems for
Lesson 11.5 are available in
blackline format in the Chapter 11
Resource Book, p. 82 and at
www.mcdougallittell.com.
56. x2 + (y + 9)2 = 100
52. DC 18 cm
53. m™DBC 68°
54. BC 7.3 cm
2
2
58. (x º 8) + (y º 2) = 11
C
68ⴗ
A
18 cm
B
WRITING EQUATIONS Write the standard equation of the circle with the
given center and radius. (Review 10.6)
57. (x + 4)2 + (y º 5)2 = 10.24 55. center (º2, º7), radius 6
56. center (0, º9), radius 10
58. center (8, 2), radius 1
1
57. center (º4, 5), radius 3.2
FINDING MEASURES Find the indicated measure. (Review 11.4)
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw a circle.
Then draw and label a sector of
that circle so that the area of the
sector is 4␲. Sample answer: a
59. Circumference
12 12 in.
៣
60. Length of AB
A
C
25π ≈ 78.5 in.
698
61. Radius
A
C
53ⴗ
13 ft
circle with radius 4 units and the
sector with the measure of the
intercepted arc 90°
2. WRITING How is the procedure
for finding the area of a sector
similar to finding arc length?
51. BD 19.4 cm
D
689π
}} ≈ 12.0 ft
180
A
C 129ⴗ
31.6 m
B
B
1896
}} ≈ 14.0 m
43π
Chapter 11 Area of Polygons and Circles
See answer at right.
Additional Test Preparation:
2. Sample answer: In both procedures you first find the
ratio of the circle represented by the sector or arc.
Then you multiply this ratio by the area of the circle to
698
find the area of the sector, or by the circumference of the
circle to find the arc length of the intercepted arc.