You are hard at work in the school science lab when your teacher asks you to turn a knob on a detector you are using degrees. Unfortunately, you have been working in radians for a while, and so you're having trouble remembering how far to turn the knob. Is there a way to translate the instructions in degrees to radians?

Read this Concept, and at the conclusion you'll be able to accomplish this task and turn the knob the appropriate amount.

### Watch This

James Sousa Example: Converting Angles in Degree Measure to Radian Measure

### Guidance

Since degrees and radians are different ways of measuring the distance moved around the circumference of a circle, it is reasonable to suppose that there is a conversion formula between these two units. This formula works for all degrees and radians. Remember that: . If you divide both sides of this equation by , you will have the conversion formula:

If we have a degree measure and wish to convert it to radians, then manipulating the equation above gives:

#### Example A

Convert to degree measure.

From the last section, you should recognize that this angle is a multiple of (or 60 degrees), so there are 11, 's in this angle, .

Here is what it would look like using the formula:

#### Example B

Convert to radian measure. Leave the answer in terms of .

and reducing to lowest terms gives us

You could also have noticed that 120 is . Since is radians, then 120 is 2, ’s, or . Make it negative and you have the answer, .

#### Example C

Express radians terms of degrees.

Note: Sometimes students have trouble remembering if it is or . It might be helpful to remember that radian measure is almost always expressed in terms of . If you want to convert from radians to degrees, you want the to cancel out when you multiply, so it must be in the denominator.

### Vocabulary

**
Radian:
**
A
**
radian
**
(abbreviated rad) is the angle created by bending the radius length around the arc of a circle.

**
Degree:
**
A
**
degree
**
is a unit for measuring angles in a circle. There are 360 of them in a circle.

### Guided Practice

1. Convert the following degree measures to radians. All answers should be in terms of .

, , , ,

2. Convert the following degree measures to radians. All answers should be in terms of .

, , , ,

3. Convert the following radian measures to degrees

, , , ,

**
Solutions:
**

1. , , , ,

2. , , , ,

3. , , , ,

### Concept Problem Solution

Since you now know that the conversion for a measurement in degrees to radians is

you can find the solution to convert to radians:

### Practice

Convert the following degree measures to radians. All answers should be in terms of .

Convert the following radian measures to degrees.

- Why do you think there are two different ways to measure angles? When do you think it might be more convenient to use radians than degrees?