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
AREAS OF CIRCLES AND SECTORS
THEOREM
THEOREM 11.7 Area of a Circle
 times the
square of the radius, or A =  r 2
The area of a circle is
r
Using the Area of a Circle
Find the area of . P.
SOLUTION
8 in.
P
Use r = 8 in the area formula.
A =  r2
=  • 82
= 64
 201.06
So, the area is 64, or about 201.06, square inches.
Using the Area of a Circle
Find the diameter of • Z.
Z
SOLUTION
Area of
•
Z = 96 cm2
The diameter is twice the radius.
A =  r2
96 =  r 2
96 = r 2

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.
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.
Using the Area of a Circle
The following theorem gives a method for finding the area of a sector.
THEOREM
THEOREM 11.8
Area of a Sector
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
r2
=
mAB
360°
, or A =
mAB
360°
• r 2
A
P
A
B
Finding the Area of a Sector
Find the area of the sector shown at the right.
C
4 ft
80°
SOLUTION
P
Sector CPD intercepts an arc whose
measure is 80°. The radius is 4 feet.
A =
=
m CD
360°
•  r2
80°
•  • 42
360°
 11.17
Write the formula for the
area of a sector.
Substitute known values.
Use a calculator.
So, the area of the sector is about 11.17 square feet.
D
Finding the Area of a Sector
Sector Area Quiz: Now it is your turn!
Find the area of the sector shown.
Ca (Central Angle) = 30°
Diameter = 16 ft
C
16 ft
SOLUTION
P
Sector CPD intercepts an arc whose
measure is 30°. The diameter is 16 feet.
Asec =
11.4
m CD
360°
30°
•  r2
Write the formula for the
area of a sector.
Circumference of Circles and Sectors
D
Finding the Area of a Sector
Sector Area Quiz: Now it is your turn!
Find the area of the sector shown at the right.
C
16 ft
30°
SOLUTION
P
Sector CPD intercepts an arc whose
measure is 30°. The diameter is 16 feet.
A =
=
m CD
360°
•  r2
30°
•  • 82
360°
 16.755
Write the formula for the
area of a sector.
Substitute known values.
Use a calculator.
So, the area of the sector is about 16.76square feet.
D
USING AREAS OF CIRCLES AND REGIONS
Finding the Area of a Region
Find the area of the shaded region shown.
5m
SOLUTION
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.
Area of
Area of
=
shaded region
circle
–
Area of
hexagon
USING AREAS OF CIRCLES AND REGIONS
Finding the Area of a Region
Area of
Area of
=
shaded region
circle
=
2
= • 5 –
= 25 –
 r2
5
2
1 •
2
75
2
3
–
Area of
hexagon
–
1
aP
2
•
(6 • 5)
5m
The apothem of a hexagon is
1 • side length •
2
3
So, the area of the shaded region is 25 – 75 3,
2
or about 13.59 square meters.
3
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.
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
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 rectangle + Area of sector – Area of circle
=
6
11
•
2
180°
+
••
360°
=
33 + 1 • 
=
33 + 9  – 4
2
•
9 – 
•
2
3
– •
1
2 •
4
(2)2
2
 34.57
The area of the front of the case is about 34.57 square inches.
2