1. 1. A circle graph has a section marked “Potatoes: 28%.”
Circles and Arcs
Lesson 10-6
Lesson Quiz
1. A circle graph has a section marked “Potatoes: 28%.”
What is the measure of the central angle of this section?
2. Explain how a major arc differs from a minor arc.
Use O for Exercises 3–6.
3. Find mYW.
4. Find mWXS.
5. Suppose that P has a diameter 2 in. greater than the diameter of
O. How much greater is its circumference? Leave your answer in terms of .
6. Find the length of XY. Leave your answer in terms of . .
100.8
A major arc is greater than a semicircle. A minor arc is smaller
than a semicircle. 30
270 2 9
10-7

2. Areas of Circles and Sectors
Lesson 10-7
Check Skills You’ll Need
(For help, go to Lesson 10-6.)
1. What is the radius of a circle with diameter 9 cm?
2. What is the diameter of a circle with radius 8 ft?
3. Find the circumference of a circle with diameter 12 in.
4. Find the circumference of a circle with radius 3 m.
The radius is half the diameter: 9 ÷ 2 = 4.5 cm
The diameter is twice the radius: (8)(2) = 16 ft
C = d =(12) = 12, or about 37.7 in.
C = 2r = 2(3) = 6, or about 18.8 m
5. Suppose that circle P has a diameter 2 in. greater than the diameter of circle O. How much greater is its circumference? Leave your answer in terms of  .
6. Find the length of XY. Leave your answer in terms of . 2 9
Check Skills You’ll Need
10-7

5. Areas of Circles and Sectors
Lesson 10-7
Notes
10-7

6. Areas of Circles and Sectors
Lesson 10-7
Notes
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 circle O.
10-7

7. Areas of Circles and Sectors
Lesson 10-7
Notes
10-7

8. Areas of Circles and Sectors
Lesson 10-7
Notes
A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle.
10-7

9. Areas of Circles and Sectors
Lesson 10-7
Notes
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.
10-7

10. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
Real-World Connection
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 your answer to the nearest whole number.
Find the areas of the archery target and the bull’s-eye.
The radius of the archery target is = 1 ft. 2
Because the diameters are in different units, convert 1 ft to 12 in.
The radius of the archery target is 1 ft = 12 in.
The area of the archery target is r2 = (12)2 = in.2
10-7

11. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
The radius of the red region is = 3 in. 6 2
(continued)
The area of the red region is r2 = (3)2 = in.2
area of archery target – area of red region = area of yellow region
– = 135
Use a calculator
The area of the yellow region is about 424 in.2
Quick Check
10-7

12. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
Finding the Area of a Sector of a Circle
Find the area of sector ACB. Leave your answer in terms of .
area of sector ACB = • r 2
mAB
360
= • (6)2
100
360
= • 36 5 18
= 10
The area of sector ACB is m2.
Quick Check
10-7

13. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
Finding the Area of a Segment of a Circle.
Find the area of the shaded segment. Round your answer to the nearest tenth.
Step 1: Find the area of sector AOB.
= • (24)2 Substitute.
120
360
= • = Simplify. 1 3
area of sector AOB = • r2 Use the formula for
area of a sector.
mAB
10-7

14. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
You can use a 30°-60°-90° triangle to find the height h of AOB and AB.
24 = 2h hypotenuse = 2 • shorter leg
12 = h Divide each side by 2.
= 3 • 12 = longer leg = 3• shorter leg
AB = Multiply each side by 2. AB 2
(continued)
Step 2: Find the area of AOB.
AOB has base ft ft, or ft and height 12 ft.
A = bh Area of a triangle
A = ( )(12) Substitute 24 for b and 12 for h.
A = Simplify. 1 2
10-7

15. Areas of Circles and Sectors
Lesson 10-7
Additional Examples
Step 3: Subtract the area of AOB from the area of sector AOB
to find the area of the segment of the circle.
(continued)
area of segment = –
Use a calculator.
To the nearest tenth, the area of the shaded segment is ft2.
Quick Check
10-7

16. Areas of Circles and Sectors
Lesson 10-7
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.
2. A circle has an 8-in. radius. Find the area of a sector whose
arc measures 135. Leave your answer in terms of .
For Exercises 3 and 4, find the area of the shaded segment.
Round to the nearest whole unit.
Lesson Quiz
1571 m2
24 in.2
15 cm2
138 in.2
10-7