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# Conversion between Degrees and Radians

## Convert between radians and degrees

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Practice Conversion between Degrees and Radians
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Most people are familiar with measuring angles in degrees. It is easy to picture angles like 30\begin{align*}30^\circ\end{align*}45\begin{align*}45^\circ\end{align*} or 90\begin{align*}90^\circ\end{align*} and the fact that 360\begin{align*}360^\circ\end{align*} makes up an entire circle. Over 2000 years ago the Babylonians used a base 60 number system and divided up a circle into 360 equal parts. This became the standard and it is how most people think of angles today.

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?

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 r\begin{align*}r\end{align*} is 2πr\begin{align*}2 \pi r\end{align*}. Just over six radii (exactly 2π\begin{align*}2 \pi\end{align*} radii) would stretch around any circle.

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

360 degrees=2π radians\begin{align*}360 \ degrees=2 \pi \ radians\end{align*}, so 180π degrees=1 radian\begin{align*}\frac{180}{\pi} \ degrees=1 \ radian\end{align*}

Alternatively; 360 degrees=2π radians\begin{align*}360 \ degrees=2 \pi \ radians\end{align*}, so 1 degree=π180 radians\begin{align*}1 \ degree= \frac{\pi}{180} \ radians\end{align*}

The conversion factor to convert degrees to radians is: π180\begin{align*}\frac{\pi}{180^\circ}\end{align*}

The conversion factor to convert radians to degrees is: 180π\begin{align*}\frac{180^\circ}{\pi}\end{align*}

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

If you were to convert 150\begin{align*}150^\circ\end{align*} into radians, you would multiply 150\begin{align*}150^\circ\end{align*} by the correct conversion factor. You would get:

150π180=15π18=5π6 radians\begin{align*}150^\circ \cdot \frac{\pi}{180^\circ}=\frac{15\pi}{18}=\frac{5 \pi}{6} \ radians\end{align*}

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

If you were to convert π6\begin{align*}\frac{\pi}{6}\end{align*} radians into degrees, you would multiply π6\begin{align*}\frac{\pi}{6}\end{align*} by the correct conversion factor. You would get π6180π=1806=30\begin{align*}\frac{\pi}{6} \cdot \frac{180^\circ}{\pi}=\frac{180^\circ}{6}=30^\circ\end{align*}

Often the π\begin{align*}\pi\end{align*}’s will cancel.

### Examples

#### Example 1

Earlier, you were asked how many radians make up a circle. Exactly 2π\begin{align*}2 \pi\end{align*} radians describe a circular arc. This is because 2π\begin{align*}2 \pi\end{align*} radii wrap around the circumference of any circle.

#### Example 2

Convert (6π)\begin{align*}(6 \pi)^\circ\end{align*} into radians.

Don’t be fooled just because this has π\begin{align*}\pi\end{align*}. This number is about 19\begin{align*}19^\circ\end{align*}

(6π)π180=6π2180=π23\begin{align*}(6 \pi)^\circ \cdot \frac{\pi}{180^\circ}=\frac{6 \pi^2}{180}=\frac{\pi^2}{3}\end{align*}

It is very unusual to ever have a π2\begin{align*}\pi^2\end{align*} term, but it can happen.

#### Example 3

Convert 5π6\begin{align*}\frac{5\pi}{6}\end{align*} into degrees.

5π6180π=5301=150\begin{align*}\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi}=\frac{5 \cdot 30^\circ}{1}=150^\circ\end{align*}

#### Example 4

Convert 210\begin{align*}210^\circ\end{align*} into radians.

INSERT 4

210π180=730π630=7π6\begin{align*}210^\circ \cdot \frac{\pi}{180^\circ}=\frac{7 \cdot 30 \cdot \pi}{6 \cdot 30}=\frac{7\pi}{6}\end{align*}

#### Example 5

Draw a π2\begin{align*}\frac{\pi}{2}\end{align*} angle by first drawing a 2π\begin{align*}2 \pi\end{align*} angle, halving it and halving the result.

π2=90\begin{align*}\frac{\pi}{2}=90^\circ\end{align*}

### Review

Find the radian measure of each angle.

1. 120\begin{align*}120^\circ\end{align*}

2. 300\begin{align*}300^\circ\end{align*}

3. 90\begin{align*}90^\circ\end{align*}

4. 330\begin{align*}330^\circ\end{align*}

5. 270\begin{align*}270^\circ\end{align*}

6. 45\begin{align*}45^\circ\end{align*}

7. (5π)\begin{align*}(5 \pi)^\circ\end{align*}

Find the degree measure of each angle.

8. 7π6\begin{align*}\frac{7 \pi}{6}\end{align*}

9. 5π4\begin{align*}\frac{5 \pi}{4}\end{align*}

10. 3π2\begin{align*}\frac{3 \pi}{2}\end{align*}

11. 5π3\begin{align*}\frac{5 \pi}{3}\end{align*}

12. π\begin{align*}\pi\end{align*}

13. π6\begin{align*}\frac{\pi}{6}\end{align*}

14. 3

15. Explain why if you are given an angle in degrees and you multiply it by π180\begin{align*}\frac{\pi}{180}\end{align*} you will get the same angle in radians.

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

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### Vocabulary Language: English

subtended arc

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