Arc Length and Radian Measure

Name ______Module 5

Learning Target: I canexpress 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 diameteris 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 in a circle of radius ?

In general, if the circumference of a circle with radius ris , then the length of an arcof degree measure ?° on a circle of radius ris .

*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 and with , calculate the measure in degrees of each arc intercepted by . Then, calculate thearc length intercepted by on each circle. The inner circle has a radius of , and each circle has a radius units greater than the previous circle.

Measure of arc in circle with radius Arc length in circle with radius

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 measureof the central angle that intercepts an arc of length equal to one radius length.

4. If 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 ?

6. What is the degree measure of an angle of radians?

7. What is the degree measure of an angle of 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?

9. In circle O, the length of radiusis 5 centimeters and the length of is 5 centimeters. What is the measure of ?

10. Ileana buys a large circular pizza that is divided into eight equal slices. She measures along the outer edge of the crust from one piece and finds it to be inches. What is the diameter of the pizza to the nearest inch?

Name ______Module 5

Arc Length and Radian MeasureProblem Set

1. and are points on the circle of radius , and the measure of arc is .

Find the length of .

2. What is the radius of a circle if the length of a arc is ?

3. and both have an angle measure of , but their arc lengths are not the same.

and . What are the arc lengths of and ?

4. In the circles shown, find the value of . Figures are not drawn to scale. The circles have central angles of equal measure.

5. Find the value of x.

6. Find the value of x.

7. Find the value of x.

8. The concentric circles all have center . The measure of the central angle is . 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 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) 2

(3)

(4) 8

Name ______Module 5

Arc Length and Radian MeasureExit Ticket

1. Find the arc length of .

2. The radius of the following circle is , and the . What is the arc length of ?