Radian Measure, Arcs and Sectors

Degrees are a human convention with $360$ arbitrary parts to a full turn. Radians, by contrast, measure angle using the circle itself: one radian is the angle subtended at the center by an arc equal in length to the radius. Because the full circumference is $2\pi r$, a complete revolution is $2\pi$ radians, which gives the master conversion

$$\pi \text{ radians}=180^\circ.$$

From this, to convert degrees to radians multiply by $\frac{\pi}{180}$, and to convert radians to degrees multiply by $\frac{180}{\pi}$. Common values worth memorizing include $30^\circ=\frac{\pi}{6}$, $45^\circ=\frac{\pi}{4}$, $60^\circ=\frac{\pi}{3}$, $90^\circ=\frac{\pi}{2}$, and $180^\circ=\pi$. Radians are not merely an alternative unit; they are essential in calculus, where the derivative rules for trigonometric