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

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

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

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

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

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

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

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 = 144 in.2 10-7

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 = 9 in.2 area of archery target – area of red region = area of yellow region 144 – 9 = 135 Use a calculator. 135 424.11501 The area of the yellow region is about 424 in.2 Quick Check 10-7

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 10 m2. Quick Check 10-7

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 = • 576 = 192 Simplify. 1 3 area of sector AOB = • r2 Use the formula for area of a sector. mAB 10-7

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 = 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. A = bh Area of a triangle A = (24 3 )(12) Substitute 24 for b and 12 for h. A = 144 3 Simplify. 1 2 10-7

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 = 192 – 144 3 353.77047 Use a calculator. To the nearest tenth, the area of the shaded segment is 353.8 ft2. Quick Check 10-7

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. 3. 4. Lesson Quiz 1571 m2 24 in.2 15 cm2 138 in.2 10-7