10-7
10-7
Areas of Circles and Sectors
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To find the areas of circles,
1. What is the radius of a circle with diameter 9 cm?
sectors, and segments of
circles
. . . And Why
To compare the area of
different-size pizzas, as in
Example 1
Lesson 10-6
4.5 cm
Objectives
1
2. What is the diameter of a circle with radius 8 ft? 16 ft
3. Find the circumference of a circle with diameter 12 in. 12π or about 37.7 in.
Examples
4. Find the circumference of a circle with radius 3 m. 6π or about 18.8 m
1
2
New Vocabulary • sector of a circle • segment of a circle
3
1
Finding Areas of Circles and Parts of Circles
Areas of circles and sectors can
be estimated without using the
constant p. One estimation
technique measures the areas
of inscribed and circumscribed
polygons much as lower and
upper sums estimate Riemann
integrals. Calculating geometric
probability may require calculating
the areas of sectors and circles.
• Use a compass to draw a large
circle. Fold the circle horizontally
and vertically. Cut the circle into
four wedges on the fold lines.
More Math Background: p. 530D
• Fold each wedge into quarters.
Cut each wedge on the fold lines.
You will have 16 wedges.
• Tape the wedges to a piece
of paper to form the figure
shown here.
Real-World Connection
Finding the Area of a Sector
of a Circle
Finding the Area of a
Segment of a Circle
Math Background
Hands-On Activity: Exploring the Area of a Circle
Lesson Planning and
Resources
See p. 530E for a list of the
resources that support this lesson.
r
PowerPoint
b
Bell Ringer Practice
1. How does the area of the figure compare with area of the circle?
They are equal.
2. The base of the figure is formed by arcs of the circle. Explain how the
length b relates to the circumference C of the circle. b N 12 C
For intervention, direct students to:
3. Explain how the length b relates to the radius r of the circle. b N πr
Finding Circumference
Check Skills You’ll Need
Lesson 10-6: Example 4
Extra Skills, Word Problems, Proof
Practice, Ch. 10
4. If you increase the number of wedges, the figure you create becomes
more and more like a rectangle with base b and height r. Write an
expression for the area of the rectangle in terms of r. πr 2
Lesson 10-7 Areas of Circles and Sectors
Special Needs
To find the areas of circles,
sectors, and segments of
circles
Below Level
L1
After Example 1, have students contact a local pizza
parlor and find out the diameters and prices of a
small and large pizza. Have them determine which is a
better buy.
learning style: tactile
575
L2
Encourage students to label parts when finding areas
within circles. For example, label the segment “I” and
the triangle “II” in Example 3.
learning style: visual
575
2. Teach
In the diagrams on the preceding page,
area of a circle = area of a “parallelogram” < b ? r < 12C ? r = pr 2,
and the approximations improve as the circle is cut into more pieces.
Guided Instruction
Key Concepts
Theorem 10-11
The area of a circle is the product of p and the
square of the radius.
Hands-On Activity
Discuss as a class why base b is
approximately but not exactly
equal to pr. Students should
mention that a segment is shorter
than a curved path between the
same two points.
1
EXAMPLE
3
EXAMPLE
Connection
area of small pizza = p(5)2 = 25p
area of medium pizza = p(6)2 = 36p
Find the radii.
Use the formula for area of a circle.
difference in area = 36p - 25p = 11p
Math Tip
< 3 4 . 557 519
Use a calculator.
There is about 35 in.2 more pizza in the medium pizza.
Quick Check
Y
1 How much more pizza is in a 14-in.-diameter pizza than in a 12-in. pizza?
about 41 in.2
A sector of a circle is a region bounded by an arc of the circle and the two radii to
the arc’s endpoints. You name a sector using one arc endpoint, the center of the circle,
and the other arc endpoint. The slice of pizza at the left is sector XOY of a circle O.
Teaching Tip
Ask: How is finding the area
of a sector like finding arc
length? Both have the ratio
measure of the arc
as a factor.
360
Real-World
radius of small pizza = 10
2 =5
radius of medium pizza = 12
2 =6
Discuss the fact that the medium
pizza has a 20 percent greater
radius but a 44 percent greater
area than the small pizza. Have
students verify these figures and
explain the difference in the
percents of increase.
2
EXAMPLE
Food How much more pizza is in a 12-in.-diameter pizza than in a 10-in. pizza?
The Activity’s method of finding
the area of a curved region is
further developed in the study
of calculus.
EXAMPLE
r
O
A = pr 2
Connection to Calculus
1
Area of a Circle
X
The area of a sector is a fractional part of the area of a circle. The ratio of a sector’s
of the arc
area to a circle’s area is measure360
.
O
Key Concepts
Theorem 10-12
Tactile Learners
Area of a Sector of a Circle
A
The area of a sector of a circle is the product of
of the arc
the ratio measure360
and the area of the circle.
Students may cut apart segments
of a circle and relate the areas to
the area of the circle.
r
O
0
2
Area of sector AOB = mAB
360 ? pr
2
EXAMPLE
B
Finding the Area of a Sector of a Circle
Find the area of sector ZOM. Leave your answer in terms of p.
0
2
area of sector ZOM = mZM
360 ? pr
72 ? p(20)2
= 360
Z
20 cm
= 80p
72⬚
M
O
The area of sector ZOM is 80p cm2.
Quick Check
576
Chapter 10 Area
Advanced Learners
576
2 Critical Thinking A circle has a diameter of 20 cm. What is the area of a sector
bounded by a 2088 major arc? Round your answer to the nearest tenth.
181.5 cm2
English Language Learners ELL
L4
Following Example 1, have students investigate how to
make a pizza with twice the area of another pizza.
Help students distinguish between segment of a circle
and line segments. Then have students compare
finding the length of an arc of a circle with finding
the area of a sector of a circle.
learning style: verbal
learning style: verbal
PowerPoint
A part of a circle bounded by an arc and the segment
joining its endpoints is a segment of a circle. To find
the area of a segment for a minor arc, draw radii to form
a sector. The area of the segment equals the area of the
sector minus the area of the triangle formed.
⫺
This pizza is cut into two
segments.
Area of sector
3
EXAMPLE
Segment
of a
circle
⫽
Area of triangle
Additional Examples
1 A circular archery target has
a 2-ft diameter. It is yellow except
for a red bull’s-eye at the center
with a 6-in. diameter. Find the area
of the yellow region. Round to the
nearest whole number. 424 in.2
2 Find the area of sector ACB.
Leave your answer in terms of p.
Area of segment
Finding the Area of a Segment of a Circle
C
Find the area of the shaded segment. Round your answer to the nearest tenth.
0
A
Use the formula
2
area of sector AOB = mAB
360 ? pr
for area of a sector.
90 ? p(10)2
= 360
Substitute.
O 10 in.
= 14 ? 100p
6 m 100°
A
10π m2
B
3 Find the area of the shaded
segment. Round your answer to
the nearest tenth.
= 25p
area of AOB = 12bh
Use the formula for area of a triangle.
= 12 (10)(10)
B
Substitute.
A
= 50
area of segment = 25p - 50
< 28.539816
120° 24 ft
O
Use a calculator.
The area of the segment is about 28.5 in.2
Quick Check
B
353.8 ft2
3 A circle has a radius of 12 cm. Find the area of the smaller segment of the circle
determined by a 608 arc. Round your answer to the nearest tenth. 13.0 cm2
Resources
• Daily Notetaking Guide 10-7
L3
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
• Daily Notetaking Guide 10-7—
L1
Adapted Instruction
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
(page 576)
Closure
Find the area of each circle. Leave your answer in terms of π.
1.
2.
3.
4.
6m
11 cm
1.7 ft
2
in.
3
π
2
0.7225π ft2
30.25π cm2
9π m2
9 in.
5. Agriculture Some farmers use a circular irrigation method. An irrigation arm
acts as the radius of an irrigation circle. How much more land is covered with
an irrigation arm of 300 ft than by an irrigation arm of 250 ft? about 86,394 ft2
6. What is the difference in the areas of a circular table with diameter 6 ft and a
circular table with diameter 8 ft? about 22 ft2
Lesson 10-7 Areas of Circles and Sectors
A circle with a diameter of 10 in.
contains an inscribed square as
shown below. Find the area of
the shaded segments. Leave your
answer in terms of p.
(25π – 50) in.2
577
577
Example 2
3. Practice
(page 576)
Find the area of each shaded sector of a circle. Leave your answer in terms of π.
7.
8.
9.
45⬚
Assignment Guide
18 yd
C Challenge
35-40
Test Prep
Mixed Review
41-43
44-48
10.
11.
30⬚ 12 in.
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 18, 26, 29, 30.
Error Prevention!
Exercises 1–4 Some students may
Example 3
substitute d for r in the formula
A = pr2. Use these exercises to
identify and correct the error.
(page 577)
12π
4 ft
17.
120⬚
18.
19.
8 ft
20. radius 15 m, arc 608 20.4 m2
B
Apply Your Skills
22. (243π ± 162) ft2
L3
L2
Name
L1
Class
Date
cm2
23.
18 ft
24.
120⬚
m2
60⬚
12 m
9 cm
25. (4 – π) ft2
L4
Practice 10-7
24. (120π ± 36 "3)
21. radius 14 cm, arc 1208 120.4 cm2
Find the area of the shaded region. Leave your answer in terms of π and in
simplest radical form. 22–27. See left.
22.
23. (54π ± 20.25 "3)
Practice
60⬚
22.1 cm2
18.3 ft2
3.3 m2
A circle has the given radius. Find the area of the smaller segment of the circle
determined by an arc with the given measure. Round to the nearest tenth.
bow of a canoe is its front end.
Adapted Practice
16 cm
56π cm2
45⬚
12π ft2
in.2
6m
Exercise 40 Point out that the
Reteaching
m2
12.
6 cm
GPS Guided Problem Solving
169π
6
Find the area of sector TOP in (O using the given information. Leave your
answer in terms of π.
0
0
2
2
14. r = 6 ft, mTP = 158 3π
13. r = 5 m, mTP = 908 25π
4 m
2 ft
0
0
15. d = 16 in., mPT = 1358 24π in.2
16. d = 15 cm, mPT = 1808
28.125π cm2
Find the area of each shaded segment. Round your answer to the nearest tenth.
Diversity
Enrichment
120⬚
64π cm2
40.5π yd2
1 A B 1-34
26 m
16 cm
L3
26. (64 – 16π) ft2
27. (784 – 196π)
in.2
25.
26.
2 ft
27.
8 ft
14 in.
8 ft
Surface Areas and Volumes of Spheres
Find the surface area of each sphere. Round your answers to the
nearest tenth.
1.
2.
4.
3.
700 m
14 in.
2 cm
5.
6.
2 ft
10 m
7m
GO
Find the volume of each sphere. Round your answers to the nearest tenth.
7.
8.
9.
12 m
14 mi
10.
11.
12.
3 cm
S.A. ⴝ 90,790 cm2
S.A. ⴝ 45,240 yd2
The volume of each sphere is given. Find the surface area. Round your
answers to the nearest whole number.
13. V = 1200 ft3
14. V = 750 m3
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-1007
© Pearson Education, Inc. All rights reserved.
40 cm
28. Multiple Choice The diver at
the right is collecting samples
from the ocean floor. The line to
the diver is 100 ft long, and the
diver is working at a depth of
80 ft. What is the approximate
area of the circle that the
diver can cover? A
11,300 ft2
25,400 ft2
31,400 ft2
51,400 ft2
100 ft
80 ft
15. V = 4500 cm3
Use the given circumference to approximate the volume of each object.
Round your answers to the nearest whole number.
16. a baseball with C = 24 cm
17. a basketball with C = 75 cm
18. a volleyball with C = 69 cm
19. a golf ball with C = 13.5 cm
578
Chapter 10 Area
29. Lower outside; the lower
inside and top pieces
have base areas 8π in.2,
but the lower outside
pieces have base areas
8.75π in.2.
578
4. Assess & Reteach
29. Writing The American Institute of Baking
GPS suggests a technique for cutting and serving
a tiered cake. The tiers of a cake have the
same height and have radii 8 in. and 13 in.
The top tier and the cake directly under it
are each cut into 8 wedges as shown. The
outer ring of the 13-inch tier is cut into 12
pieces. Which would be larger, a piece from
the top or a piece from the outer ring? Explain.
PowerPoint
Lesson Quiz
Solve.
See margin, p. 578.
30. How many circles with radius 4 in. will have the same total area as a circle with
radius 12 in.? 9 circles
31. Games A dart board has diameter 20 in. and is divided into 20 congruent
sectors. Find the area of one sector. Round your answer to the nearest tenth.
15.7 in.2
32. In a circle, a 908 sector has area 36p in.2. What is the circle’s radius? 12 in.
Real-World
Connection
Careers Training as a pastry
chef is part of the background
of many executive chefs.
C
Challenge
38. Blue region;
let AB ≠ 2x. Area of
blue ≠ 4x 2 – πx2;
area of yellow ≠
πx 2 – 2x 2.
2
39. 200π
3 – 50"3 units
Problem Solving Hint
For Exercise 39, where
must T and U lie in a
diagram of (T and (U?
th
t pa
40a. Boa
10 ft
33. Open-Ended Draw a circle and a sector so that the area of the sector is
16p cm2. Give the radius of the circle and the measure of the arc of the sector.
See margin.
34. A method for finding the area of a segment determined by a minor arc is
described on page 577. 34a. See margin.
a. Describe
0 two ways to find the area of a segment determined by a major arc.
b. If mAB = 90 in
a circle of radius 10, find the areas of the two segments
0
determined by AB . 25π – 50; 75π ± 50
2. A circle has an 8-in. radius. Find
the area of a sector whose arc
measures 135. Leave your
answer in
terms of p. 24π in.2
Find the area of the shaded region. Leave your answer in terms of π.
3.
35.
36.
60° 13 cm
10 m
7m
4m
(200 – 50π) m2
(49π – 73.5"3) m2
38. Circle O at the right is inscribed in square ABCD and
circumscribed about square PQRS. Which is smaller,
the blue region or the yellow region? Explain.
39. Circles T and U each have a radius 10 and TU = 10. Find
the area of the region that is contained inside both circles.
4π m2
A
B
Q
P
4.
22 in.
S
R
D
C
40. Recreation An 8 ft-by-10 ft floating dock is anchored
in the middle of a pond. The bow of a canoe is tied to a
corner of the dock with a 10-ft rope as shown in the picture below.
a. Sketch a diagram of the region in which the bow of the canoe can travel.
b. Write a plan for finding the area. 40a–b. See left.
c. Find the area. Round your answer to the nearest square foot. 239 ft2
138 in.2
Alternative Assessment
Have student partners plan how
to sell pizzas. Using the area
formulas from this lesson, they
should select two different sizes
and price them proportionately.
Have students write a paragraph
explaining how the size of each
pizza justifies its price.
2 ft
40b. Find the area of 34 of a
circle of radius 10 and add
1
4 of a circle of radius 2.
Lesson 10-7 Areas of Circles and Sectors
lesson quiz, PHSchool.com, Web Code: aua-1007
33. Answers may vary.
Sample: 8 cm radius;
15 cm2
O
8 ft
10 ft
For Exercises 3 and 4, find the
area of the shaded segment.
Round to the nearest whole unit.
37.
10 ft
8 ft
1. A park contains two circular
playgrounds. One has a
diameter of 60 m, and the
other has a diameter of
40 m. How much greater
is the area of the larger
playground? Round to the
nearest whole number.
1571 m2
908 arc
8 cm
579
34. a. Answers may vary.
Sample: Subtract the
minor arc segment
area from the area of
the circle, or add the
areas of the major
sector and k formed.
579
Test Prep
Test Prep
Resources
Multiple Choice
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 593
• Test-Taking Strategies, p. 588
• Test-Taking Strategies with
Transparencies
Use this Checkpoint Quiz to check
students’ understanding of the
skills and concepts of Lessons 10-4
through 10-7.
Resources
41. A circle has area 72p yd2. What is the area of a 10° sector of the circle? A
A. 2p yd2
B. 3p yd2
C. 4p yd2
D. 6p yd2
42. A sector of 90° has area p mm. What is the area of the circle? G
F. 2p mm2
G. 4p mm2
H. 8p mm2
J. 4 mm2
Short Response
43. Each of three water sprinklers covers a
12 m
semicircle of radius 2 m. The shaded
2m
region remains dry. Find the area of
the shaded region to the nearest square
meter. Show your work or explain how you found the area. See margin.
Mixed Review
GO for
Help
Lesson 10-6
0
Find the length of AB in each circle. Leave your answers in terms of π.
B
44. A
45.
46. A
Grab & Go
• Checkpoint Quiz 2
60⬚
30 cm
Lesson 6-5
Lesson 4-7
O
B
72⬚
A
5m
36 in.
O
B
40⬚
28π in.
10π cm
2π m
47. Three sides of a trapezoid are congruent. The fourth side is 4 in. longer than
each of the other three. The perimeter is 49 in. Find the length of each side.
1114 in., 1114 in., 1114 in., 1514 in.
48. Write a proof. (Hint: First prove
B
D
overlapping triangles are congruent.)
C
F
Given: AC > BC, &A > &B See margin.
E
Prove: #BDF > #AEF
A
Checkpoint Quiz 2
Lessons 10-4 through 10-7
The similarity ratio of kABC to kDEF is 3 : 5. Fill in the missing information.
1. The perimeter of #ABC is 36 in.
The perimeter of #DEF is 9. 60 in.
43. [2] area of rectangle ≠
2 ? 12 ≠ 24
area of semicircles ≠
3 ? 12π? 22 ≠ 6π
24 – 6π N 5.2 m2
(
3. The areas of two similar triangles are 1.44 and 1.00. Find their similarity ratio.
6:5
4. Find the area of #ABC if a = 4 ft, b = 7 ft and m&C = 20. 4.8 ft2
)
[1] no work shown
48. Show that kBCE O
kACD by ASA. Then use
CE O CD and segment
subtraction to show that
BD O AE. Now with
vertical lBFD O
lAFE, kBDF O kAEF
by ASA.
5. A regular 12-gon has perimeter 24 cm. Find its area to the nearest tenth. 44.8 cm2
6. 100π in.2
7. 27π m2
8. (16π – 32) cm2
9. 22.5π ft2
Find the area of each shaded region. Leave answers in terms of π.
6.
7.
20 in.
8.
120⬚
9m
9.
8 cm
6 ft
225⬚
0
0
10. In a circle of radius 18 mm, mAB = 45. Find the length of AB in terms of p. 9π
2 mm
580
580
2. The area of #ABC is 9. 45 in.2
The area of #DEF is 125 in.2.
Chapter 10 Area