Arc Lengths and Areas of Sectors
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Transcript of Arc Lengths and Areas of Sectors
Arc Lengths and Areas of SectorsArc Lengths and Areas of Sectors
Arc LengthArc Length
The length of part of the The length of part of the circumference. 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 circleSector of a circle
A region bounded by 2 radii and an A region bounded by 2 radii and an arc.arc.
.
Portions of a Circle: Determine the Arc measure based on the portion given.
A. B. C. D.
¼ of a circle: ½ of a circle: 1/3 of circumference : 6π out of a total 36π on the circle: ¼ ● 360
90o
90o
½ ● 360
180o
180o
1/3 ● 360
120o
120o
1/6 ● 360
60o
60o
Area of a Sector Area of a Sector FormulaFormula
m360πr2Area of a sector =
measure of the central angle or arc
The fraction of the circle!
The area of the entire circle!
.
Arc Length FormulaArc Length Formula
m3602πrArc Length =
measure of the central angle or arc
The fraction of the circle!
The circumference of the entire circle!
.
90
6
m AOB
radius
240
12
m AOB
radius
300
12
m AOB
radius
120
2.4
m AOB
radius
108
10 2
m AOB
radius
Find the length of AB and the area of sector AOB.
1. 2. 3. 4. 5.
Length of AB Length of AB Length of AB Length of AB Length of AB
Area of sector AOB Area of sector AOB Area of sector AOB Area of sector AOB Area of sector AOB
Fraction of circle:
¼
90o
90o
6
Fraction ● circumference
¼ ● 12π
3π units
Fraction ● area
¼ ● 36π
9π units2
240o
240o
12
Fraction of circle:
2/3
Fraction ● circumference
2/3 ● 24π
16π units
Fraction ● area
2/3 ● 144π
96π units2
300o
300o 12
Fraction of circle:
5/6
5/6 ● 24π
20π units
Fraction ● area
5/6 ● 144π
120π units2
A
AB
B
O OAO
B
120o
120o
2.4AO
B
Fraction of circle:
1/3
Fraction ● circumference
1/3 ● 4.8π
1.6π units
Fraction ● area
1/3 ● 5.76π
1.92π units2
108o
108o
10√2A
B
O
Fraction of circle:
3/10
3/10 ● 20√2π
6√2π units
Fraction ● area
3/10 ● 200π
60π units2
6. The area of sector AOB is 48π and 270m AOB . Find the radius of ○O.
m
360πr2Area of a sector =
270
360πr248π =
3
4r248 =
4
3
4
3
16
r264 =
r = 8
9
4 40m AOB 7. The area of sector AOB is and . Find the radius of ○O.
m
360πr2Area of a sector =
40
360πr2 π =
9
41
9r2 =
9
4
9
1
9
1
r2 =81
4
r = 9
2
60˚
8 612
60 430
OO
O
8. 9. 11.
Find the area of the shaded region. Point O marks the center of the circle.
10.
160
3π units2 9π - 18 units2 24π - 36√3 units2 8π - 8√3 units2
Some common fractions and Some common fractions and measures!measures!
Arc or Central Arc or Central Angle MeasureAngle Measure
Fraction of the Fraction of the CircleCircle
Arc or Central Arc or Central Angle MeasureAngle Measure
Fraction of the Fraction of the CircleCircle
3636oo 108108oo
1/61/6 5/65/6
120120oo 2/32/3
3030oo 11/1211/12
1/81/8 5/85/8
1/10
1/3
1/12
3/10
60o
45o
300o
240o
225o
330o