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ARC LENGTH AND SECTOR AREA
Section 4.1A
Precalculus PreAP/Dual, Revised ©2017
§4.1A: Arc Length and Sector Area 18/1/2018 12:09 AM
ARC LENGTH
How do we find the arc length? What is circumference of a circle?
§4.1A: Arc Length and Sector Area 2
𝑻
𝑺
𝑹centralangle
( )2C r=
2C r
=
S r
S r
=
=S r=8/1/2018 12:09 AM
ARC LENGTH
A. Sector is a region of the circle that bounded by two radii and an arc of a circle
B. The Central Angle of a sector is the angle formed by the two radii
C. Arc Length equation: 𝒔 = 𝒓𝜽
D. Degrees must be converted to Radians
E. Do NOT forget the units
§4.1A: Arc Length and Sector Area 38/1/2018 12:09 AM
EXAMPLE 1
Determine the Arc Length of a circle with the given radius of 𝒓 = 𝟒
inches and 𝜽 =𝝅
𝟔.
§4.1A: Arc Length and Sector Area 4
s r=
( )46
s
=
Arc Length
2
3s in
=
8/1/2018 12:09 AM
EXAMPLE 2
The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds? 1 full rotation is 60 seconds. Round to 4 decimal places.
§4.1A: Arc Length and Sector Area 5
s r=
( )15
260
=
Arc Length
( )minutes
2hour
=
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EXAMPLE 2
The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds? 1 full rotation is 60 seconds. Round to 4 decimal places.
§4.1A: Arc Length and Sector Area 6
9.4247s in=
2
=
s r=
( )62
s
=
3s =
8/1/2018 12:09 AM
YOUR TURN
Determine the Arc Length of a circle with the given radius of 𝒓 = 𝟒inches and 𝜽 = 𝟐𝟒𝟎°. Make sure to convert degrees to radians.
§4.1A: Arc Length and Sector Area 7
s r=
( ) ( )4 240180
s
=
16
3s in
=
( )4
43
s
=
8/1/2018 12:09 AM
AREA OF A SECTOR AREA
A. Sector is a circle of the region bounded by two radii of the circle and their intercepted arc
B. Equation: 𝑨 =𝟏
𝟐𝒓𝟐𝜽 where radius is the base and 𝜽 is the height
C. Always leave answers in radian mode
§4.1A: Arc Length and Sector Area 88/1/2018 12:09 AM
AREA OF A SECTOR
§4.1A: Arc Length and Sector Area 9
21
2A r =
2C r=
2
360A r
=
2
2A r
=
21
2A r=
8/1/2018 12:09 AM
EXAMPLE 3
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 𝟏𝟐𝟎°. Find the area of the fairway watered by the sprinkler.
§4.1A: Arc Length and Sector Area 10
120 =
( )2
120180 3
= =
21
2A r =
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EXAMPLE 3
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 𝟏𝟐𝟎°. Find the area of the fairway watered by the sprinkler.
§4.1A: Arc Length and Sector Area 11
24900
3feet
=
( )2
120180 3
= =
( )21 2
702 3
A
=
8/1/2018 12:09 AM
EXAMPLE 4
A sector of a circle of radius 24 m has an area of 288 𝒎². Find the central angle of this sector.
§4.1A: Arc Length and Sector Area 12
1 radian =8/1/2018 12:09 AM
EXAMPLE 5
Three circles with radii 1, 2, and 3 feet are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.
§4.1A: Arc Length and Sector Area 13
2
4A ft
=
2
131
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YOUR TURN
A sector of a circle has a central angle of 𝟔𝟎°. Find the area of the sector if the radius of the circle is 3 miles.
§4.1A: Arc Length and Sector Area 14
23
2miles
8/1/2018 12:09 AM
ASSIGNMENT
Worksheet
§4.1A: Arc Length and Sector Area 158/1/2018 12:09 AM