1. Arc Length and Sector Area
Section 4.1A
Precalculus PreAP/Dual, Revised ©2017
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§4.1A: Arc Length and Sector Area

2. §4.1A: Arc Length and Sector Area
How do we find the arc length? What is circumference of a circle? 𝑻 𝑺 𝑹
central
angle
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§4.1A: Arc Length and Sector Area

3. §4.1A: Arc Length and Sector Area
Sector is a region of the circle that bounded by two radii and an arc of a circle
The Central Angle of a sector is the angle formed by the two radii
Arc Length equation: 𝒔=𝒓𝜽
Degrees must be converted to Radians
Do NOT forget the units
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§4.1A: Arc Length and Sector Area

4. §4.1A: Arc Length and Sector Area
Example 1
Determine the Arc Length of a circle with the given radius of 𝒓=𝟒 inches and 𝜽= 𝝅 𝟔 .
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§4.1A: Arc Length and Sector Area

5. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

6. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

7. §4.1A: Arc Length and Sector Area
Your Turn
Determine the Arc Length of a circle with the given radius of 𝒓=𝟒 inches and 𝜽=𝟐𝟒𝟎°. Make sure to convert degrees to radians.
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§4.1A: Arc Length and Sector Area

8. §4.1A: Arc Length and Sector Area
Area of a Sector Area
Sector is a circle of the region bounded by two radii of the circle and their intercepted arc
Equation: 𝑨= 𝟏 𝟐 𝒓 𝟐 𝜽 where radius is the base and 𝜽 is the height
Always leave answers in radian mode
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§4.1A: Arc Length and Sector Area

9. §4.1A: Arc Length and Sector Area
Area of a Sector
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§4.1A: Arc Length and Sector Area

10. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

11. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

12. §4.1A: Arc Length and Sector Area
Example 4
A sector of a circle of radius 24 m has an area of 288 𝒎². Find the central angle of this sector.
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§4.1A: Arc Length and Sector Area

13. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

14. §4.1A: Arc Length and Sector Area
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.
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§4.1A: Arc Length and Sector Area

15. §4.1A: Arc Length and Sector Area
Assignment
Worksheet
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§4.1A: Arc Length and Sector Area