1. Areas of Circles and Sectors
Section 11.3
Areas of Circles and Sectors

2. 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.

4. 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 or 88 inches. Divide by 2 to find that the radius is 44 inches.
A = πr2 Area of a circle
= π(44)2 Substitution
≈ Use a calculator
The area of the cover is about 6082 square inches.

5. 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.

6. 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.

7. 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.

8. 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
≈ in.2 Use a calculator.
The area of one slice of pie is about square inches.

9. 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.

10. 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
≈ in.2 Use a calculator.