Radian to Degree Measure

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.  In radians, one complete counterclockwise revolution is $2\pi$ and in degrees, one complete counterclockwise revolution is $360°$ . So, degree measure and radian measure are related by the equations

$360°=2\pi$ radians and

$180°=\pi$ radians

From the latter, we obtain the equation $1$ radian = ${\left(\frac{180}{\pi }\right)}^{\text{o}}$ .  This leads us to the rule to convert radian measure to degree measure.  To convert from radians to degrees, multiply the radians by $\frac{180°}{\pi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{radians}}$ .

Example 1:

Convert $\frac{\pi }{4}$ radians to degrees.

$\left(\frac{\pi }{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}\right)\left(\frac{180°}{\pi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}}\right)={\left(\frac{180}{4}\right)}^{\text{o}}=45°$

Example 2:

Convert $\frac{9\pi }{5}$ radians to degrees.

$\left(\frac{9\pi }{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}\right)\left(\frac{180°}{\pi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}}\right)=9{\left(36\right)}^{\text{o}}=324°$

Example 3:

Convert $3$ radians to degrees.

$\left(3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}\right)\left(\frac{180°}{\pi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{rad}}\right)={\left(\frac{540}{\pi }\right)}^{\text{o}}\approx 171.89°$