11.5 Areas of Circles and Sectors

1
11.5 Areas of Circles and Sectors

• Geometry
• Mrs. Spitz
• Spring 2005

2
Objectives/Assignment

• Find the area of a circle and a sector of a
  circle.
• Use areas of circles and sectors to solve
  real-life problems such as finding the areas of
  portions of circles.
• Assignment pp. 695-696 1-34 all

3
Areas of Circles and Sectors

• The diagrams on the next slide show regular
  polygons inscribed in circles with radius r.
  Exercise 42 on pg. 697 demonstrates that as the
  number of sides increases, the area of the
  polygon approaches the value ?r2.

4
Examples of regular polygons inscribed in circles.
4 -gon
3 -gon
5 -gon
6 -gon
5
Thm. 11.7 Area of a Circle

• The area of a circle is ? times the square of the
  radius or A ?r2.

6
Ex. 1 Using the Area of a Circle

• Find the area of P.

• Solution
• Use r 8 in the area formula.
• A ?r2
• ? 82
• 64?
• ? 201.06

• So, the area if 64?, or about 201.06 square
  inches.

7
Ex. 1 Using the Area of a Circle

• Find the diameter of
• Z.

• Solution
• Area of circle Z is 96 cm2.
• A ?r2
• 96 ? r2
• 96 r2
• ?
• 30.56 ? r2
• 5.53 ? r

• The diameter of the circle is about 11.06 cm.

8
More . . .

• A sector of a circle is a region bounded by two
  radii of the circle and their intercepted arc.
  In the diagram, sector APB is bounded by AP, BP,
  and . The following theorem gives a method
  for finding the area of a sector.

9
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.

m
m
A

or
A

?r2
?r2
360
360
10
Ex. 2 Finding the area of a sector

• Find the area of the sector shown below.

Sector CPD intercepts an arc whose measure is
80. The radius is 4 ft.
m
A

?r2
360
11
Ex. 2 Solution
Write the formula for area of a sector.
Substitute known values.
Use a calculator.
? 11.17

• So, the area of the sector is about 11.17 square
  feet.

12
Ex. 3 Finding the Area of a Sector

• A and B are two points on a P with radius 9
  inches and m?APB 60. Find the areas of the
  sectors formed by ?APB.

FIRST draw a diagram of P and ?APB. Shade
the sectors. LABEL point Q on the major arc.
FIND the measures of the minor and major arcs.
60
13
Ex. 3 Finding the Area of a Sector

• Because m?APB 60, m 60 and
• m 360 - 60 300.

Use the formula for the area of a sector.
Area of small sector
60

? 92
360
1

? 81
6
? 42.41 square inches
14
Ex. 3 Finding the Area of a Sector

• Because m?APB 60, m 60 and
• m 360 - 60 300.

Use the formula for the area of a sector.
300
Area of large sector

?r2
360
60

? 92
360
5

? 81
6
? 212.06 square inches
15
Using Areas of Circles and regions

• You may need to divide a figure into different
  regions to find its area. The regions may be
  polygons, circles, or sectors. To find the area
  of the entire figure, add or subtract the areas
  of separate regions as appropriate.

16
Ex. 4 Find the Area of a Region

• Find the area of the region shown.

The diagram shows a regular hexagon inscribed in
a circle with a radius of 5 meters. The shaded
region is the part of the circle that is outside
the hexagon.
Area of Circle
Area of Hexagon
Area of Shaded

17
Solution
Area of Circle
Area of Hexagon
Area of Shaded


?r2
½ aP
(? 52 ) ½ ( v3) (6 5)
75
25? - v3, or about 13.59 square meters.
2
18
Ex. 5 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?

19
Ex. 5 Finding the Area of a Region
20
More . . .

• Complicated shapes may involve a number of
  regions. In example 6, the curved region is a
  portion of a ring whose edges are formed by
  concentric circles. Notice that the area of a
  portion of the ring is the difference of the
  areas of the two sectors.

21
Upcoming

• 11.6 Geometric Probability
• Chapter 11 Review
• Chapter 11 Test
• Chapter 12 Definitions
• Chapter 12 Postulates/Theorems