Most people are familiar with measuring angles in degrees. It is easy to picture angles like

However, there are many units with which to measure angles. For example, the gradian was invented along with the metric system and it divides a circle into 400 equal parts. The sizes of these different units are very arbitrary.

A radian is a unit of measuring angles that is based on the properties of circles. This makes it more meaningful than gradians or degrees. How many radians make up a circle?

### Radians and Degrees

A **radian** is defined to be the central angle where the subtended **arc length** is the same length as the radius.

Another way to think about radians is through the circumference of a circle. The circumference of a circle with radius

To define a radian in terms of degrees, equate a circle measured in degrees to a circle measured in radians.

Alternatively;

The conversion factor to convert degrees to radians is:

The conversion factor to convert radians to degrees is:

If an angle has no units, it is assumed to be in radians.

If you were to convert

** 150∘⋅π180∘=15π18=5π6 radians**

You can check your work by making sure the degree units cancel.

If you were to convert ** π6⋅180∘π=180∘6=30∘**

Often the

### Examples

#### Example 1

Earlier, you were asked how many radians make up a circle. Exactly

#### Example 2

Convert

Don’t be fooled just because this has

It is very unusual to ever have a

#### Example 3

Convert

#### Example 4

Convert

INSERT 4

#### Example 5

Draw a

### Review

Find the radian measure of each angle.

1.

2.

3.

4.

5.

6.

7.

Find the degree measure of each angle.

8.

9.

10.

11.

12.

13.

14. 3

15. Explain why if you are given an angle in degrees and you multiply it by

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.1.