CIR - Degrees and Radians Lesson

Degrees and Radians

You are likely very familiar with measuring angles in degrees. However, there is another unit of angle measure that exists, a radian.

What is a radian? A radian is the measure of the central angle of a circle subtended by an arc of equal length to the radius. Watch the video to see how to mark off three radians on a circle.

How many radians are in a circle? In any circle, there are approximately 6.28 radians or, more specifically, $LaTeX: 2\pi$ $2\pi$ radians.

What is the relationship between degrees and radians?

**SPECIAL NOTE** In the video below, we are asked to find AG and it should say, find the length of arc AG.

Convert from degrees to radians: Multiply by equation image indicator $LaTeX: \frac{\pi}{180}$ $\frac{\pi}{180}$

Convert from radians to degrees: Multiply by equation image indicator $LaTeX: \frac{180}{\pi}$ $\frac{180}{\pi}$

Example

Given angle equation image indicator $LaTeX: \theta=135^\circ$ $\theta=135^\circ$ , we can convert it to radians by multiplying by $LaTeX: \frac{\pi}{180}$ $\frac{\pi}{180}$ .

equation image indicator $LaTeX: 135\left(\frac{\pi}{180}\right)=\frac{135\pi}{180}=\frac{3\pi}{4}$ $135\left(\frac{\pi}{180}\right)=\frac{135\pi}{180}=\frac{3\pi}{4}$

Given angle equation image indicator $LaTeX: \theta=\frac{\pi}{2}$ $\theta=\frac{\pi}{2}$ , we can convert it to degrees by multiplying by $LaTeX: \frac{180}{\pi}$ $\frac{180}{\pi}$ .

equation image indicator $LaTeX: \frac{\pi}{2}\left(\frac{180}{\pi}\right)=\frac{180\pi}{2\pi}=90^\circ$ $\frac{\pi}{2}\left(\frac{180}{\pi}\right)=\frac{180\pi}{2\pi}=90^\circ$

Important Fact: If an angle is in degrees, there will be a degree symbol. If an angle is in radians there will be no symbol!

Let's check your understanding. Match the angles below:

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