Name ______________________________
Arc Length and Radian Measure
Module 5
Learning Target: I can express the measure of an angle in radians and calculate the length of an arc.
Opening Exercise
1. What is the circumference of a circle with radius 10 cm.
2. A circular pizza with a 12 inch diameter is cut into eight congruent pieces. What is the measure of
the central angle in each slice?
Arc Length and Radian Measure
In the opening exercise we calculated the circumference of a circle with radius 10 cm. Recall that the
entire arc around a circle measures 360°. Now let’s calculate the length of a fraction of that arc.
1. What is the length of the arc of degree that measures 60° in a circle of radius 10 cm?
In general, if the circumference of a circle with radius r is 2𝜋𝑟, then the length of an arc of degree
𝑥
measure 𝒙° on a circle of radius r is (360) 2𝜋𝑟.
*Notice that there is a difference between the measure of an arc in degrees and the length of an arc.
2. Given the concentric circles with center 𝐴 and with 𝑚∠𝐴 = 60°, calculate the measure in degrees
of each arc intercepted by ∠𝐴. Then, calculate the arc length intercepted by ∠𝐴 on each circle. The
inner circle has a radius of 10, and each circle has a radius 10 units greater than the previous circle.
̅̅̅̅ = _______
Measure of arc in circle with radius 𝐴𝐵
̅̅̅̅ = _______
Arc length in circle with radius 𝐴𝐵
Measure of arc in circle with radius ̅̅̅̅
𝐴𝐶 = _______
Arc length in circle with radius ̅̅̅̅
𝐴𝐶 = _______
̅̅̅̅ = _______
Measure of arc in circle with radius 𝐴𝐷
̅̅̅̅ = _______
Arc length in circle with radius 𝐴𝐷
3. What is the arc length if the central angle has a measure of 1°?
Note: Mathematicians have used this relationship to define a new angle measure, a radian. A radian
is the measure of the central angle that intercepts an arc of length equal to one radius length.
𝜋
4. If 1° = 180 radians, what does 180° equal in radian measure? What does 360° equal in radian
measure?
1 radian is approximately 57.3 degrees.
5. What is the radian measure of an angle of 27°?
6. What is the degree measure of an angle of −
3𝜋
4
radians?
7. What is the degree measure of an angle of 2𝜋 radians?
8. What is the length of an arc intercepted by a central angle of 60o in a circle with a radius of 7
inches?
̅̅̅̅ is 5 centimeters and the length of 𝐴𝐵
̂ is 5 centimeters. What is
9. In circle O, the length of radius 𝑂𝐵
the measure of ∠𝐴𝑂𝐵?
10. Ileana buys a large circular pizza that is divided into eight equal slices. She measures along the
1
outer edge of the crust from one piece and finds it to be 5 2 inches. What is the diameter of the pizza
to the nearest inch?
Name ______________________________
Arc Length and Radian Measure
Module 5
Problem Set
̂ is 72°.
1. 𝑃 and 𝑄 are points on the circle of radius 5 cm, and the measure of arc 𝑃𝑄
̂.
Find the length of 𝑃𝑄
2. What is the radius of a circle if the length of a 45° arc is 9𝜋?
̂ and 𝐶𝐷
̂ both have an angle measure of 30°, but their arc lengths are not the same.
3. 𝐴𝐵
̅̅̅̅ = 4 and 𝐵𝐷
̂ and 𝐶𝐷
̂?
̅̅̅̅ = 2. What are the arc lengths of 𝐴𝐵
𝑂𝐵
4. In the circles shown, find the value of 𝑥. Figures are not drawn to scale. The circles have central
angles of equal measure.
5. Find the value of x.
𝑥
6. Find the value of x.
7. Find the value of x.
8. The concentric circles all have center 𝐴. The measure of the central angle is 45°. The arc lengths
are given. Find the radius of each circle.
9. A sprinkler system is set up to water the sector shown in the accompanying diagram, with angle
ABC measuring 1 radian and radius 𝐴𝐵 = 20 feet. What is the length of arc AC, in feet?
10. A dog has a 20-foot leash attached to the corner where a garage and a fence meet, as shown in
the accompanying diagram. When the dog pulls the leash tight and walks from the fence to the
garage, the arc the leash makes is 55.8 feet. What is the measure of angle between the garage and
the fence, in radians?
11. A central angle of a circular garden measures 2.5 radians and intercepts an arc of 20 feet. What
is the radius of the garden?
12. A circle has a radius of 4 inches. In inches, what is the length of the arc intercepted by a central
angle of 2 radians?
(1)
(2)
(3)
(4)
2𝜋
2
8𝜋
8
Name ______________________________
Arc Length and Radian Measure
Module 5
Exit Ticket
̂.
1. Find the arc length of 𝑃𝑄𝑅
̂?
2. The radius of the following circle is 36 cm, and the 𝑚∠𝐴𝐵𝐶 = 60°. What is the arc length of 𝐴𝐶