1. AREAS OF CIRCLES AND SECTORS
These regular polygons, inscribed in circles with radius r, demonstrate that as the number of sides increases, the area of the polygon approaches the value  r 2.
3-gon
4-gon
5-gon
6-gon

2. AREAS OF CIRCLES AND SECTORS
THEOREM r
THEOREM Area of a Circle
The area of a circle is  times the square of the radius, or A =  r 2

3.  201.06 . Find the area of P. Use r = 8 in the area formula.
Using the Area of a Circle
Find the area of P. . P
8 in.
SOLUTION
Use r = 8 in the area formula.
A =  r 2
=  • 8 2
= 64  
So, the area is 64, or about , square inches.

4.  Find the diameter of Z. The diameter is twice the radius. A =  r 2
Using the Area of a Circle Z
Find the diameter of Z. •
SOLUTION
Area of  Z = 96 cm2 •
The diameter is twice the radius.
A =  r 2
96 =  r 2
= r 2 96 
30.56  r 2
5.53  r
Find the square roots.
The diameter of the circle is about 2(5.53), or about 11.06, centimeters.

5. In the diagram, sector APB is bounded by AP, BP, and AB.
Using the Area of a Circle
The sector of a circle is the region bounded by two radii of the circle and their intercepted arc. A P r B
In the diagram, sector APB is bounded by AP, BP, and AB.

6. Using the Area of a Circle
The following theorem gives a method for finding the area of a sector.
THEOREM
THEOREM Area of a Sector A B P
The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360°. A
 r 2 =
, or A =
• r 2
mAB
360°

7.  11.17 Find the area of the sector shown at the right.
Finding the Area of a Sector
Find the area of the sector shown at the right. P C D
4 ft
80°
SOLUTION
Sector CPD intercepts an arc whose measure is 80°. The radius is 4 feet.
m CD
360°
A =
•  r 2
Write the formula for the area of a sector.
80°
360° =
•  • 4 2
Substitute known values.
 11.17
Use a calculator.
So, the area of the sector is about square feet.

8. USING AREAS OF CIRCLES AND REGIONS
Finding the Area of a Region
Find the area of a the shaded region shown.
5 m
The diagram shows a regular hexagon inscribed in a circle with radius 5 meters. The shaded region is the part of the circle that is outside of the hexagon.
SOLUTION
Area of shaded region =
Area of circle
Area of hexagon –

9. 3 3 3  r 2  • 5 2 – USING AREAS OF CIRCLES AND REGIONS = – = – 1 2
Finding the Area of a Region
5 m
Area of shaded region =
Area of circle
Area of hexagon – =
 r 2 – 1 2
a P
The apothem of a hexagon is • side length • 1 2 3 =
 • – 1 2 5 3 •
• (6 • 5) =
25 – 3 75 2
or about square meters.
So, the area of the shaded region is 25  – 75 2 3 ,

10. Complicated shapes may involve a number of regions.
Finding the Area of a Region
Complicated shapes may involve a number of regions. P P
Notice that the area of a portion of the ring is the difference of the areas of two sectors.

11. Finding the Area of a Region
WOODWORKING You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case?
SOLUTION
The front of the case is formed by a rectangle and a sector, with a circle removed. Note that the intercepted arc of the sector is a semicircle.
Area =
Area of rectangle +
Area of sector –
Area of circle

12. 6 • 33 + •  • 9 –  • (2)2 33 +  – 4  •  34.57 = + – = + – = =
Finding the Area of a Region
WOODWORKING You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case?
Area
Area of circle
Area of rectangle = +
Area of sector –
6 •
11 2 = + –
180° 360°
•  • 32
 • 1 2 • =
•  • 9 –  • (2)2 1 2 =
33 +  – 4 9 2
 34.57
The area of the front of the case is about square inches.