Arc Length

Show your work for each problem.

1.
What is the arc length of a full circle with radius cm?
1. a) cm
2. b) cm
3. c) cm
4. d) cm
Answer: cm
The arc length of a full circle equals the circumference: cm.
2.
Which formula is used to find arc length when the angle is in radians?
1. a)
2. b)
3. c)
4. d)
Answer:
When the angle is in radians, we use . This is simpler because one radian is defined as the angle where arc length equals the radius.
3.
A semicircle has a radius of cm. What is the arc length? Give your answer in terms of .
Answer: 6pi
A semicircle has a central angle of , which is half of . Arc length = cm.
4.
Convert to radians. Give your answer in terms of .
Answer: pi/2
radians.
5.
Find the arc length of a sector with central angle and radius cm.
1. a) cm
2. b) cm
3. c) cm
4. d) cm
Answer: cm
cm.
6.
A circle has radius m. Find the arc length for a central angle of radians. Give your answer in terms of .
Answer: 2pi
Using m.
7.
Find the arc length of a sector with central angle and radius cm.
Answer: 6pi
1. What fraction of the full circle is ? (Simplify the fraction) 1/3
2. What is the circumference of the full circle ()? 18pi
3. Multiply the fraction by the circumference to get arc length 6pi
8.
The minute hand of a clock is cm long. How far does the tip travel in minutes?
Answer: 10pi
1. What fraction of an hour is 20 minutes? 1/3
2. What angle does the minute hand cover in 20 minutes? 120
3. Calculate the arc length using 10pi
9.
If the arc length is cm and the central angle is , what is the radius?
1. a) cm
2. b) cm
3. c) cm
4. d) cm
Answer: cm
. So . Therefore cm.
10.
An arc has length m and the radius is m. Find the central angle in degrees.
Answer: 90
.
11.
A circular running track has an inner radius of m and an outer radius of m. Find the difference in arc length for one semicircular end between the inner and outer lanes.
Answer: 2pi
1. What is the arc length of the outer semicircle ( m)? 32pi
2. What is the arc length of the inner semicircle ( m)? 30pi
3. What is the difference? 2pi
12.
A pizza has a radius of cm. If a slice has an arc length of cm along the crust, what is the central angle of the slice in degrees?
Answer: 90
. The pizza slice is a quarter of the pizza!
13.
A car travels around a circular curve with radius m. If the car covers an arc of radians, how far does it travel? Then convert this angle to degrees.
Answer: 100pi/3
1. Use . What is the arc length? 100pi/3
2. Convert radians to degrees 120
3. Approximate the arc length in meters (use ) 105
14.
Two circles have radii and . If both have arcs with the same central angle, what is the ratio of their arc lengths?
1. a)
2. b)
3. c)
4. d)
Answer:
From , if is constant, then is directly proportional to . So .
15.
A sector has an arc length of cm and a central angle of radians. Find the radius, then calculate the area of the sector.
Answer: 96pi
1. Use to find the radius. What is ? 16
2. Convert the angle to degrees 135
3. Calculate the sector area using (in terms of pi) 96pi