# 2.2: Conversion between Degrees and Radians

**At Grade**Created by: CK-12

**Practice**Conversion between Degrees and Radians

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

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

#### Example A

Convert

From the last section, you should recognize that this angle is a multiple of

Here is what it would look like using the formula:

#### Example B

Convert

and reducing to lowest terms gives us

You could also have noticed that 120 is

#### Example C

Express

Note: Sometimes students have trouble remembering if it is

### 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. \begin{align*}90^\circ\end{align*}, \begin{align*}396^\circ\end{align*}, \begin{align*}120^\circ\end{align*}, \begin{align*}540^\circ\end{align*}, \begin{align*}630^\circ\end{align*}

### Concept Problem Solution

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

\begin{align*}\text{degrees} \times \frac{\pi}{180}=\text{radians}\end{align*}

you can find the solution to convert \begin{align*}75^\circ\end{align*} to radians:

\begin{align*}75^\circ \times \frac{\pi}{180}= \frac{75\pi}{180} = \frac{5\pi}{12}\end{align*}

### Practice

Convert the following degree measures to radians. All answers should be in terms of \begin{align*}\pi\end{align*}.

- \begin{align*}90^\circ\end{align*}
- \begin{align*}360^\circ\end{align*}
- \begin{align*}50^\circ\end{align*}
- \begin{align*}110^\circ\end{align*}
- \begin{align*}495^\circ\end{align*}
- \begin{align*}-85^\circ\end{align*}
- \begin{align*}-120^\circ\end{align*}

Convert the following radian measures to degrees.

- \begin{align*}\frac{5\pi}{12}\end{align*}
- \begin{align*}\frac{3\pi}{5}\end{align*}
- \begin{align*}\frac{8\pi}{15}\end{align*}
- \begin{align*}\frac{7\pi}{10}\end{align*}
- \begin{align*}\frac{5\pi}{2}\end{align*}
- \begin{align*}3\pi\end{align*}
- \begin{align*}\frac{7\pi}{2}\end{align*}
- 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?

Degree

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

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

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

Here you'll learn how to convert degrees to radians, and vice versa.