11-4
Area of Circles
MAIN IDEA
Find the areas of circles.
• Fold a paper plate in half four times to
New Vocabulary
• Label the radius r as shown. Let C
sector
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• Extra Examples
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• Self-Check Quiz
divide it into 16 equal-sized sections.
1
C
2
r
represent the circumference of the circle.
• Cut out each section; reassemble to
form a parallelogram-shaped figure.
1. What is the measurement
_
of the base and the height? 1 C ; r
2
r
(height)
2. Substitute these values into
1
C (base)
2
the formula for the area of a
parallelogram. A = 1 C (r )
_
2
3. Replace C with the expression for the circumference of a circle, 2π r.
Simplify the equation and describe what it represents.
_
3. A = 1 (2πr) (r); A = πr 2; the area of a circle
2
In the Mini Lab, the formula for the area of a parallelogram was used to
develop a formula for the area of a circle.
Area of a Circle
Words
Key Concept
The area A of a circle equals the
product of π and the square of
its radius r.
Symbols
Model
r
A = πr 2
Find the Area of a Circle
1 Find the area of the circle.
A = πr 2
Area of a circle
A = π · 22
Replace r with 2.
[π]
2
2 in.
12.56637061
The area of the circle is approximately 12.6 square inches.
a. Find the area of a circle with a radius of 3.2 centimeters. Round to
the nearest tenth. 32.2 cm 2
Lesson 11-4 Area of Circles
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2 COINS Find the area of the face of the Wisconsin quarter shown.
The diameter of the quarter is 24 millimeters,
1( )
so the radius is _
24 or 12 millimeters.
24 mm
2
A = πr 2
Area of a circle
A = π · 12 2
Replace r with 12.
A ≈ 452.4
Use a calculator.
The area is approximately 452.4 square
millimeters.
b. POOLS The bottom of a circular swimming pool with diameter
30 feet is painted blue. How many square feet are blue? 706.5 ft 2
Radii
The plural form of radius
is radii.
A sector of a circle is a region of a circle bounded by two radii.
3 Ellis draws a circle with a diameter of
16 inches, and shades one region of the
circle. Find the approximate area of
the sector.
16 in.
A 100 in 2
C 402 in 2
180°
B 201 in 2
D 804 in 2
Read the Item
Identifying What is
Given Before finding
area, be sure to read
the question carefully
and identify if the
radius or diameter
is given.
The diameter of the circle is 16 inches. Since there are 360° in
1 the area of the entire circle.
180°
a circle, the sector is _
or _
360°
2
Solve the Item
A = πr 2
A=π·8
Area of a circle
2
A ≈ 200
Replace r with 16 ÷ 2 or 8.
Multiply. Use 3.14 for π.
1
The area of the sector is approximately _
(200) or 100 square inches.
2
The answer is A.
c. Ray drew one circle with a radius of 7 centimeters and another
circle with a radius of 10 centimeters. Find the approximate
difference between the areas of the circles. H
F 28 cm 2
590
G 40 cm 2
H 160 cm 2
J 254 cm 2
Chapter 11 Measurement: Two- and Three-Dimensional Figures
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★ indicates multi-step problem
Examples 1, 2
Find the area of each circle. Round to the nearest tenth.
(pp. 589–590)
78.5 cm
1.
5 cm
2
254.5 in
2.
2
4. diameter = 13 ft
9 in.
132.7 ft 2
Example 3
5. MULTIPLE CHOICE Kenneth draws the circle
(p. 590)
shown at the right. He shades one region
of the circle. What is the approximate area
of the sector? B
A 88 yd 2
201.1 m 2
3. diameter = 16 m
14 yd
C 310 yd 2
B 154 yd 2
D 615 yd 2
Answers were computed using the π key on a calculator.
HOMEWORK
For
Exercises
6–7, 10–11,
14–15, 19
8–9, 12–13,
16–18
36–38
HELP
See
Examples
Find the area of each circle. Round to the nearest tenth.
201.1 cm 2 7.
6.
28.3 in 2
3 in.
95.0 ft 2
8.
11 ft
1
8 cm
2
3
Exercise Levels
A: 6–19
B: 20–29
C: 30–35
227.0 cm 2 10.
9.
17 cm
12. diameter = 8.4 m
55.4 m
2
15. radius = 3_ ft
2
3
4
2.4 m
18.1 m 2
3.2 mm
13. diameter = 12.6 cm
124.7 cm
2
16. diameter = 9_ mi
2
32.2 mm 2
11.
1
4
14. radius = 4_ in.
1
2
63.6 in 2
17. diameter = 20_ yd
44.2 ft
67.2 mi
18. PATCHES Find the area of the Girl Scout patch
shown if the diameter is 1.25 inches. Round
to the nearest tenth. 1.2 in 2
338.2 yd
2
3
4
19. TOOLS A sprinkler that sprays water in a
circular area can be adjusted to spray up to
30 feet. To the nearest tenth, what is the
maximum area of lawn that can be watered
by the sprinkler? 2,827.4 ft 2
ESTIMATION Estimate to find the approximate area of each circle.
20.
8 cm
Sample
answer:
3 × 42 =
48 cm 2
21.
5.9 ft
Sample
22.
answer:
3 × 62 =
108 ft 2
13.8 in.
Sample
answer:
3 × 72 =
147 in 2
Lesson 11-4 Area of Circles
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For Exercises 23–26, use a compass to draw the circle
shown on centimeter grid paper.
23. Count the number of squares that lie completely
within the circle. Then count the number of squares
that lie completely within or contain the circle.
24. Estimate the area of the circle by finding the mean
of the two values you found in Exercise 23.
25. Find the area of the circle by using the area formula.
26. How do the areas you found in Exercises 24 and 25 compare to one another?
27. A semicircle is half a circle. Find the area of the
semicircle to the nearest tenth.
8.6 m
28. Which has a greater area, a triangle with a base
of 100 feet and a height of 100 feet or a circle with
diameter of 100 feet? Justify your selection.
EXTRA
PRACTICE
29. RADIO SIGNALS A radio station sends a signal in a circular area with an 80-
mile radius. Find the approximate area in square kilometers that receives
the signal. (Hint: 1 square mile ≈ 2.6 square kilometers)
See pages 698, 714.
H.O.T. Problems
30. REASONING If the length of the radius of a
circle is tripled, does the area also triple?
Explain your reasoning.
4 ft
12 ft
CHALLENGE Find the area of the shaded region in each figure. Round
to the nearest tenth.
31.
32.
33.
8m
3.5 cm
12 m
5.25 in.
12.5 cm
34. FIND THE ERROR Dasan and Carmen are finding the area of a circle
that has a diameter of 16 centimeters. Who is correct? Explain.
A = π(16)2
≈ 804 cm2
A = π(8)2
≈ 201 cm2
Dasan
35.
Carmen
WR ITING IN MATH Write and solve a real-world problem in which you
would solve the problem by finding the area of a circle.
592
Chapter 11 Measurement: Two- and Three-Dimensional Figures
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36. The radius of the half dollar in
38. Which two figures have the same area
shaded? A
centimeters is given below. Find
the approximate area of the shaded
sector. A
8m
1.95 cm
7.5 m
12 m
12 m
Figure II
Figure I
10 m
12 m
12 m
180°
6m
A 6 cm 2
C 14 cm 2
B 12 cm 2
D 28 cm 2
A Figure I and Figure IV
37. Which equation could be used to find
the area in square inches of a circle
with a radius of 12 inches? J
F A=6×π
B Figure I and Figure II
C Figure II and Figure IV
D Figure II and Figure III
H A = 12 × π
G A = π × 62 J
Figure IV
Figure III
A = π × 12 2
39. MEASUREMENT What is the circumference of a circle that has a radius of
8 yards? Use 3.14 for π and round to the nearest tenth if necessary.
(Lesson 11-3) 50.2 yd
40. MEASUREMENT Find the area of a triangle with a base of 21 meters and
a height of 27 meters.
(Lesson 11-2)
283.5 m 2
Find the area of each parallelogram. Round to the nearest tenth if necessary.
(Lesson 11-1)
41.
39.5 cm 2
42.
5 cm
10 in.
120 in 2
12 in.
44. 8.5 2
72.25
45. 3.14 · 6 2 113.04 46.
8.7 m
7.9 cm
PREREQUISITE SKILL Simplify each expression.
100.1 m 2
43.
11.5 m
(Lessons 1-2, 1-3, and 1-4)
_1 · 5.4 2 + 11 25.58 47. _1 · 7 2 + (9)(14) 150.5
2
2
Lesson 11-4 Area of Circles
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