Section 2.2 Arc Length and Sector Area
Section 2.2 ? Arc Length and Sector Area
Arc Length
Definition
If a central angle, in a circle of a radius r, cuts off an arc of length s, then the measure of, in radians is:
r
s r
r
s r
( in radians)
Note: When applying the formula s r , the value of must be in radian.
Example
A central angle in a circle of radius 3 cm cuts off an arc of length 6 cm. What is the radian measure of .
Solution
s r
6 3
cm cm
2 rad
11
Example
A circle has radius 18.20 cm. Find the length of the arc intercepted by a
central angle with measure
3 8
radians.
Solution
Given:
3 rad, 8
r 18.20 cm
s r
18.20
3 8
cm
21.44 cm
Example The minute hand of a clock is 1.2 cm long. To two significant digits, how far does the tip of the minute hand move in 20 minutes? Solution
Given: r = 1.2 cm One complete rotation = 1 hour = 60 minutes = 2
2
20 60
20 2 60
2 3
s r
1.2
2 3
2.5 cm
12
Example
A person standing on the earth notices that a 747 jet flying overhead subtends an angle 0.45. If the length of the jet is 230 ft., find its altitude to the nearest thousand feet.
Solution
s r
r
s
230
0.45
180
230(180) 0.45
29,000 ft
Example
A rope is being wound around a drum with radius 0.8725 ft. How much rope will be wound around the drum if the drum is rotated through an angle of 39.72?
Solution
s r
0.8725
39.72
180
0.6049 feet
13
Area of a Sector
A sector of a circle is a portion of the interior of a circle intercepted by a central angle.
Area of sector
Area of circle
A r2
r2
2
r2
A
1 2
r 2
A r2
2
Central angle One full rotation
Definition
If (in radians) is a central angle in a circle with radius r, then the area of the sector formed by an angle is given by
A
1 2
r2
( in radians)
Example Find the area of the sector formed by a central angle of 1.4 radians in a circle of radius 2.1 meters Solution
Given: r = 2.1 m = 1.4
A 1 r 2 2
1 2
(2.1)2 (1.4)
3.1m2
14
Example
If the sector formed by a central angle of 15 has an area of cm2, find the radius of a circle. 3
Solution
Given:
15
180
12
A
3
A 1 r 2 2
1r2 3 2 12
24
3
1 2
r
2
12
24
8 r2
r 8
r 2 2 cm
Example A lawn sprinkler located at the corner of a yard is set to rotate 90 and project water out 30.0 ft. To three significant digits, what area of lawn is watered by the sprinkler?
Solution Given: 90 2 r 30 ft
A 1 r 2 2
1 (30)2
2
2
707 ft 2
15
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