<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Conversion between Degrees and Radians

## Convert between radians and degrees

0%
Progress
Practice Conversion between Degrees and Radians
Progress
0%

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 $75^\circ$ 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.

### 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: $\pi \ \text{radians} = 180^\circ$ . If you divide both sides of this equation by $\pi$ , you will have the conversion formula:

$\text{radians} \times \frac{180}{\pi}=\text{degrees}$

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

$\text{degrees} \times \frac{\pi}{180}=\text{radians}$

#### Example A

Convert $\frac{11\pi}{3}$ to degree measure.

From the last section, you should recognize that this angle is a multiple of $\frac{\pi}{3}$ (or 60 degrees), so there are 11, $\frac{\pi}{3}$ 's in this angle, $\frac{\pi}{3} \times 11 = 60^\circ \times 11 = 660^\circ$ .

Here is what it would look like using the formula:

$\text{radians} \times \frac{180}{\pi} = \text{degrees}$

#### Example B

Convert $-120^\circ$ to radian measure. Leave the answer in terms of $\pi$ .

$\text{degrees} \times \frac{\pi}{180}&=\text{radians}\\-120^\circ \times \frac{\pi}{180}&=\frac{-120^\circ \pi}{180}$

and reducing to lowest terms gives us $-\frac{2\pi}{3}$

You could also have noticed that 120 is $2 \times 60$ . Since $60^\circ$ is $\frac{\pi}{3}$ radians, then 120 is 2, $\frac{\pi}{3}$ ’s, or $\frac{2\pi}{3}$ . Make it negative and you have the answer, $-\frac{2\pi}{3}$ .

#### Example C

Express $\frac{11\pi}{12}$ radians terms of degrees.

$\text{radians} \times \frac{180}{\pi}=\text{degrees}$

Note: Sometimes students have trouble remembering if it is $\frac{180}{\pi}$ or $\frac{\pi}{180}$ . It might be helpful to remember that radian measure is almost always expressed in terms of $\pi$ . If you want to convert from radians to degrees, you want the $\pi$ to cancel out when you multiply, so it must be in the denominator.

### Vocabulary

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 $\pi$ .

$240^\circ$ , $270^\circ$ , $315^\circ$ , $-210^\circ$ , $120^\circ$

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

$15^\circ$ , $-450^\circ$ , $72^\circ$ , $720^\circ$ , $330^\circ$

3. Convert the following radian measures to degrees

$\frac{\pi}{2}$ , $\frac{11\pi}{5}$ , $\frac{2\pi}{3}$ , $5\pi$ , $\frac{7\pi}{2}$

Solutions:

1. $\frac{4\pi}{3}$ , $\frac{3\pi}{2}$ , $\frac{7\pi}{4}$ , $-\frac{7\pi}{6}$ , $\frac{2\pi}{3}$

2. $\frac{\pi}{12}$ , $-\frac{5\pi}{2}$ , $\frac{\pi}{5}$ , $4 \pi$ , $\frac{11\pi}{6}$

3. $90^\circ$ , $396^\circ$ , $120^\circ$ , $540^\circ$ , $630^\circ$

### Concept Problem Solution

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

$\text{degrees} \times \frac{\pi}{180}=\text{radians}$

you can find the solution to convert $75^\circ$ to radians:

$75^\circ \times \frac{\pi}{180}= \frac{75\pi}{180} = \frac{5\pi}{12}$

### Practice

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

1. $90^\circ$
2. $360^\circ$
3. $50^\circ$
4. $110^\circ$
5. $495^\circ$
6. $-85^\circ$
7. $-120^\circ$

Convert the following radian measures to degrees.

1. $\frac{5\pi}{12}$
2. $\frac{3\pi}{5}$
3. $\frac{8\pi}{15}$
4. $\frac{7\pi}{10}$
5. $\frac{5\pi}{2}$
6. $3\pi$
7. $\frac{7\pi}{2}$
8. 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?

### Vocabulary Language: English

Degree

Degree

A degree is a unit for measuring angles in a circle. There are 360 degrees in a circle.

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.
subtended arc

subtended arc

A subtended arc is the part of the circle in between the two rays that make the central angle.