Feature Lesson Geometry Lesson Main rcle graph has a section marked “Potatoes: 28%.” t is the measure of the central angle of this section? ain how a major arc differs from a minor arc. O for Exercises 3–6. mYW. mWXS. ose that P has a diameter 2 in. greater than the diameter of O. How much greater is its circumference? Leave your answer in ter the length of XY. Leave your answer in terms of . . . . major arc is greater than a semicircle. A minor arc is smaller an a semicircle. 100.8 270 30 2 9 Lesson 10-6 Circles and Arcs Lesson Quiz 10-7
Transcript
FeatureLesson
GeometryGeometry
LessonMain
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 .
.
..
A major arc is greater than a semicircle. A minor arc is smaller than a semicircle.
100.8
270
30
2
9
Lesson 10-6
Circles and ArcsCircles and Arcs
Lesson Quiz
10-7
FeatureLesson
GeometryGeometry
LessonMain
(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.
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Check Skills You’ll Need
Check Skills You’ll Need
10-7
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 = 2r = 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
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Notes
10-7
FeatureLesson
GeometryGeometry
LessonMain
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Notes
10-7
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.
FeatureLesson
GeometryGeometry
LessonMain
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Notes
10-7
FeatureLesson
GeometryGeometry
LessonMain
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Notes
10-7
A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle.
FeatureLesson
GeometryGeometry
LessonMain
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Notes
10-7
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.
FeatureLesson
GeometryGeometry
LessonMain
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 = 144 in.2
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.22
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Additional Examples
10-7
Real-World Connection
FeatureLesson
GeometryGeometry
LessonMain
The area of the red region is r2 = (3)2 = 9 in.2
The radius of the red region is = 3 in.62
(continued)
Use a calculator. 135 424.11501
The area of the yellow region is about 424 in.2
area of archery target – area of red region = area of yellow region
144 – 9 = 135
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Quick Check
Additional Examples
10-7
FeatureLesson
GeometryGeometry
LessonMain
The area of sector ACB is 10 m2.
= 10
= • 36 5 18
= • (6)2100360
Find the area of sector ACB. Leave your answer in terms of .
area of sector ACB = • r 2mAB 360
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Quick Check
Additional Examples
10-7
Finding the Area of a Sector of a Circle
FeatureLesson
GeometryGeometry
LessonMain
= • (24)2 Substitute.120360
= • 576 = 192 Simplify.13
area of sector AOB = • r2 Use the formula for area of a sector.
mAB 360
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Find the area of the shaded segment. Round your answer to
the nearest tenth.
Step 1: Find the area of sector AOB.
Additional Examples
10-7
Finding the Area of a Segment of a Circle.
FeatureLesson
GeometryGeometry
LessonMain
A = bh Area of a triangle
A = (24 3 )(12) Substitute 24 for b and 12 for h.
A = 144 3 Simplify.
1212
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 = 12 3 longer leg = 3• shorter leg
AB = 24 3 Multiply each side by 2.
AB 2
(continued)
Step 2: Find the area of AOB.
AOB has base 12 3 ft + 12 3 ft, or 24 3 ft and height 12 ft.
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Additional Examples
10-7
FeatureLesson
GeometryGeometry
LessonMain
area of segment = 192 – 144 3
To the nearest tenth, the area of the shaded segment is 353.8 ft2.
353.77047 Use a calculator.
Step 3: Subtract the area of AOB from the area of sector AOB to find the area of the segment of the circle.
(continued)
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
Quick Check
Additional Examples
10-7
FeatureLesson
GeometryGeometry
LessonMain
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.3. 4.
1571 m2
24 in.2
15 cm2 138 in.2
Lesson 10-7
Areas of Circles and SectorsAreas of Circles and Sectors
