1. Adapted from Walch Education
Defining Radians
Adapted from Walch Education

2. Key Concepts
Arc length is the distance between the endpoints of an arc, and is commonly written as
The radian measure of a central angle is the ratio of the length of the arc intercepted by the angle to the radius of the circle.
3.4.1: Defining Radians

3. Key Concepts, continued
The definition of radian measure leads us to a formula for the radian measure of a central angle  in terms of the intercepted arc length, s, and the radius of the circle,
3.4.1: Defining Radians

4. Key Concepts, continued
When the intercepted arc is equal in length to the radius of the circle, the central angle measures 1 radian.
3.4.1: Defining Radians

5. Key Concepts, continued
Recall that the circumference, or the distance around a circle, is given by or , where C represents circumference, r represents radius, and d represents the circle’s diameter.
Since the ratio of the arc length of the entire circle to the radius of the circle
is there are radians in a full
circle.
3.4.1: Defining Radians

6. Key Concepts, continued
We know that a circle contains 360° or radians. We can convert between radian measure and degree measure by simplifying this ratio to get radians = 180°.
To convert between radian measure and degree measure, set up a proportion.
3.4.1: Defining Radians

7. Key Concepts, continued
To find arc length s when the central angle is given in degrees, we determine the fraction of the circle that we want to find using the measure of the angle. Set up a proportion with the circumference, C.
3.4.1: Defining Radians

8. Practice
A circle has a diameter of 20 feet. Find the length of an arc intercepted by a central angle measuring 36°.
3.4.1: Defining Radians

9. Solution Find the circumference of the circle. Set up a proportion.
3.4.1: Defining Radians

10. Solution, continued The length of the arc is approximately 6.28 feet.
Multiply both sides by to find the arc length.
The length of the arc is approximately feet.
3.4.1: Defining Radians

11. Can you…
Convert 40° to radians.
3.4.1: Defining Radians

12. Ms. Dambreville
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