Recall Area of a Circle A = r2 Topic 11-3 Circles and Sectors Objectives: Find the areas of circles, sectors, and segments of circles Recall Area of a Circle A = r2

Real-world and Pizza 10-inch pizza 12-inch pizza How much more pizza is in a 12 inch diameter pizza than in a 10 inch diameter pizza? 10-inch pizza Radius = 10/2 = 5 Area = (5)2 = 25 12-inch pizza Radius = 12/2 = 6 Area = (6)2 = 36 Difference in Area = 36 - 25 = 11  34.6 in2

Sectors of Circles Sector of a Circle: region bounded by an arc of the circle and the two radii to the arcs endpoints name by using one arc endpoint, the center of the circle, and the other arc endpoint (AOB of Circle O) A arc radius B O radius

Area and Sectors of Circles – (Foldable) The area of a sector is a fractional part of the area of a circle. The ratio of a sector’s area to a circle’s area is: measure of the arc 360

Areas of Sectors - foldable Find the area of the shaded sector. . . area of shaded sector = 70 360 𝑥 𝜋 9 2 = • (81) 70 360 = 63 4 𝜋 ≈49.48

Areas of Sectors - example Find the area of sector ACB. Leave your answer in terms of π . . . area of sector ACB = • r 2 mAB 360 = • (6)2 100 360 = • 36 5 18 = 10 The area of sector ACB is 10 m2.

Segments of a Circle Part of the circle bounded by an arc and the segments joining the arc’s endpoints Segment of a circle

Finding Area of a Segment of a Circle (foldable) - = - ? =

Areas of segments- foldable Find the area of the shaded segment. = • (6)2 Substitute. 90 360 = 9𝜋 area of sector AOB = • r2 Use the formula for area of sector. mAB Step 1: Find the area of sector AOB. A Step 2: Find the area of ∆AOB. O B Area of ∆ = 1 2 𝑏ℎ = 1 2 6∗6=18 Step 3: Subtract area of ∆AOB from area of sector AOB Area of sector AOB = 9𝝅 - 18

Areas of segments - example 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

(continued) Step 2: Find the area of AOB. 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 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

(continued) Step 3: Subtract the area of AOB from the area of sector AOB to find the area of the segment of the circle. 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.