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.

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