Areas of Circles and Sectors Section 11.3 Areas of Circles and Sectors

In Lesson 10-1, you learned that the formula for the circumference C of a circle with radius r is given by C = 2πr. You can use this formula to develop the formula for the area of a circle.   Below, a circle with radius r and circumference C has been divided into congruent pieces and then rearranged to form a figure that resembles a parallelogram. As the number of congruent pieces increases, the rearranged figure more closely approaches a parallelogram. The base of the parallelogram is 0.5C and the height is r, so its area is 0.5C • r. Since C = 2πr, the area of the parallelogram is also 0.5(2πr)r or πr2.

The area of the cover is about 6082 square inches. Example 1: An outdoor accessories company manufactures circular covers for outdoor umbrellas. If the cover is 8 inches longer than the umbrella on each side, find the area of the cover in square inches. The diameter of the umbrella is 72 inches, and the cover must extend 8 inches in each direction. So, the diameter of the cover is 8 + 72 + 8 or 88 inches. Divide by 2 to find that the radius is 44 inches. A = πr2 Area of a circle = π(44)2 Substitution ≈ 6082.1 Use a calculator The area of the cover is about 6082 square inches.

Example 2: Find the radius of a circle with an area of 58 square inches. A = πr2 Area of a circle 58 = πr2 Substitution Divide each side by π. Take the positive square root of each side. 4.3 ≈ r Simplify The radius of the circle is about 4.3 in.

A slice of a circular pizza is an example of a sector of a circle A slice of a circular pizza is an example of a sector of a circle. A sector of a circle is a region of a circle bounded by a central angle and its intercepted major or minor arc. The formula for the area of a sector is similar to the formula for arc length.

Example 3:   a) A pie has a diameter of 9 inches and is cut into 10 congruent slices. What is the area of one slice to the nearest hundredth? Find the arc measure of a pie slice. Since the pie is equally divided into 10 slices, each slice will have an arc measure of 360 ÷ 10 or 36. Find the radius of the pie. Use this measure to find the area of the sector, or slice. The diameter is 9 inches, so the radius is 4.5 inches. Area of a sector x = 36 and r = 4.5 ≈ 6.36 in.2 Use a calculator. The area of one slice of pie is about 6.36 square inches.

Example 3:   b) A pizza has a diameter of 14 inches and is cut into 8 congruent slices. What is the area of one slice to the nearest hundredth? Find the arc measure of a pie slice. Since the pie is equally divided into 8 slices, each slice will have an arc measure of 360 ÷ 8 or 45. Find the radius of the pie. Use this measure to find the area of the sector, or slice. The diameter is 14 inches, so the radius is 7 inches. Area of a sector x = 45 and r = 7 ≈ 19.24 in.2 Use a calculator. The area of one slice of pie is about 19.24 square inches.

Example 4: Find the area of the shaded sector Example 4: Find the area of the shaded sector. Round to the nearest tenth. a) Area of a sector x = 53 and r = 4 ≈ 7.4 ft2 Use a calculator.

Example 4: Find the area of the shaded sector Example 4: Find the area of the shaded sector. Round to the nearest tenth. b) Area of a sector x = 295 and r = 11 ≈ 311.5 in.2 Use a calculator.