Transcript 11.5 Areas of Circles and Sectors

11.5 Areas of Circles and
Sectors
Geometry
Mrs. Spitz
Spring 2006
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
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.
Examples of regular polygons
inscribed in circles.
3 -gon
5 -gon
4 -gon
6 -gon
Theorem. 11.7: Area of a Circle
• The area of a circle is  times the
square of the radius or A = r2.
r
Ex. 1: Using the Area of a Circle
• Find the area of
8 in.
P
P.
Solution:
a. Use r = 8 in the
area formula.
A = r2
=  • 82
= 64
 201.06
So, the area if 64, or about 201.06 square inches.
Ex. 1: Using the Area of a Circle
• Find the diameter of
Z.
Z
Solution:
b. 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.
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.
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 = m
r2
360°
or
A =
m
360°
• r2
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.
C
4 ft.
A =
P
D
m
360°
• r2
Ex. 2 Solution
A =
m
360°
= 80°
360°
 11.17
• r2
Write the formula for
area of a sector.
• r2
Substitute known values.
Use a calculator.
So, the area of the sector is about
11.17 square feet.
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.
Q
A
P
60°
9
B
FIRST draw a diagram of
and APB. Shade the
sectors.
LABEL point Q on the major
arc.
FIND the measures of the
minor and major arcs.
P
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° • r2
360°
= 60° •  • 92
360°
= 1
•  • 81
6
 42.41 square inches
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 large sector = 300° • r2
360°
= 60° •  • 92
360°
= 5
•  • 81
6
 212.06 square inches
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.
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
Shaded
=
Area of
Area of
Circle
Hexagon
Solution:
Area of
Shaded
=
Area of
Area of
Circle
Hexagon
r2
½ aP
=
= ( • 52 ) – ½ • (
75
2
√3) • (6 • 5)
= 25 √3, or about 13.59
square meters.
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?
Ex. 5: Finding the Area of a Region
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.
Upcoming:
•
•
•
•
•
11.6 Geometric Probability
Chapter 11 Review before Chapter 11 Test
Chapter 12 Definitions
Chapter 12 Postulates/Theorems
Begin Reviewing for Final Exam – You must
take and pass the final exam to pass the
class!!!
• Absences: More than 10, I will fail you per
attendance policy.