Transcript 11.6 Arc Lengths and Area of Sectors

Geometry
11.6 Arc Lengths and Areas of Sectors
Arc Length

The length of part of the circumference.
The length of the arc depends on what two things?
1) The measure of the arc.
2) The size of the circle.
An arc length measures distance while
the measure of an arc is in degrees.
Sector of a circle

A region bounded by 2 radii and an arc.
.
 Portions of a Circle: Determine the Arc measure based on the portion given.
180o
90o
90o
120o
180o
60o
120o
60o
A.
B.
C.
D.
¼ of a circle:
¼ ● 360
½ of a circle:
½ ● 360
1/3 of circumference :
6π out of a total 36π
on the circle:
1/3 ● 360
1/6 ● 360
Area of a Sector Formula
measure of the central angle or arc
Area of a sector =
m
2
πr
360
The area of the entire circle!
The fraction of the circle!
.
Arc Length Formula
measure of the central angle or arc
m
2πr
360
The circumference of the
entire circle!
Arc Length =
The fraction of the circle!
.
Find the length of AB and the area of sector AOB.
1.
A
90o
O
90o
6
2.
240o
3.
240o
B
O
12
A
300o
120o
4. B
300o 12
A
O
O
120o
2.4
5.
A
B
108o
108o
O 10√2
B
B
m  AO B  90
m  A O B  240
m  AO B  300
m  A O B  120
m  A O B  108
radius  6
radius  12
radius  12
radius  2.4
radius  10 2
Fraction of circle:
Fraction of circle:
Fraction of circle:
Fraction of circle:
2/3
5/6
1/3
Fraction of circle:
¼
Length of AB
Fraction ● circumference
¼ ● 12π
3π units
Area of sector AOB
Fraction ● area
Length of AB
Fraction ● circumference
Length of AB
2/3 ● 24π
16π units
5/6 ● 24π
20π units
Area of sector AOB
Area of sector AOB
Fraction ● area
Fraction ● area
Length of AB
Fraction ● circumference
1/3 ● 4.8π
1.6π units
Area of sector AOB
Fraction ● area
3/10
Length of AB
3/10 ● 20√2π
6√2π units
Area of sector AOB
Fraction ● area
¼ ● 36π
2/3 ● 144π
5/6 ● 144π
1/3 ● 5.76π
3/10 ● 200π
9π units2
96π units2
120π units2
1.92π units2
60π units2
A
6. The area of sector AOB is 48π and m  AOB  270. Find the radius of ○O.
Area of a sector =
m
πr2
360
270
48π =
πr2
360
4 16 3 2 4
48 =
r
3
4
3
64 =
r2
r=8
7. The area of sector AOB is
Area of a sector =
9
 and m  AOB  40 . Find the radius of ○O.
4
m
πr2
360
9
40
π=
πr2
4
360
99 1 2 9
=
r
1 4 9
1
81 2
= r
4
r= 9
2
Find the area of the shaded region. Point O marks the center of the circle.
8.
9.
11.
10.
60˚
8
160
3
π units2
12
O
6
9π - 18 units2
30
O
60
24π - 36√3 units2
O
4
8π - 8√3 units2
HW

P. 453-454 WE #1-19
Some common fractions and measures!
Arc or Central Angle
Measure
Fraction of the Circle
Arc or Central Angle
Measure
Fraction of the Circle
36o
1/10
108o
3/10
60o
1/6
300o
5/6
120o
1/3
240o
2/3
30o
1/12
330o
11/12
45o
1/8
225o
5/8