Conversion Formulas

Since the circumference of a circle is $2\pi r,$ the radian measure corresponding to $360^\circ$ is easily found with the preceding formula

$$\theta=\frac{s}{r}\;\implies\;\theta=\frac{2\pi r}{r}\;\implies\;\theta=2\pi.$$

Notice that the radius $r$ cancels (and hence the units in which it is measured,) leaving the radian measure of an angle as a ``unitless'' number.

It follows that an angle with degree measure $180^\circ$ corresponds to a radian measure of $\pi.$ Thus, we have the proportion:

$$\frac{\theta^\circ}{180^\circ}=\frac{\theta \text{rad}}{\pi \text{rad}}$$

from which follow the conversion formulas:

$$= \left(\frac{\pi}{180}\right)$$ radians

$$= \left(\frac{180}{\pi}\right).$$ degrees