### SubsectionB.2.1Angles — Radians vs Degrees

For mathematics, and especially in calculus, it is much better to measure angles in units called radians rather than degrees. By definition, an arc of length $\theta$ on a circle of radius one subtends an angle of $\theta$ radians at the centre of the circle. The circle on the left has radius 1, and the arc swept out by an angle of $\theta$ radians has length $\theta\text{.}$ Because a circle of radius one has circumference $2\pi$ we have

\begin{align*} 2\pi\text{ radians }&=360^\circ & \pi\text{ radians }&=180^\circ & \frac{\pi}{2}\text{ radians }&=90^\circ\\ \frac{\pi}{3}\text{ radians }&=60^\circ & \frac{\pi}{4}\text{ radians }&=45^\circ & \frac{\pi}{6}\text{ radians }&=30^\circ \end{align*}

More generally, consider a circle of radius $r\text{.}$ Let $L(\theta)$ denote the length of the arc swept out by an angle of $\theta$ radians and let $A(\theta)$ denote the area of the sector (or wedge) swept out by the same angle. Since the angle sweeps out the fraction $\frac{\theta}{2\pi}$ of a whole circle, we have

\begin{align*} L(\theta) &= 2\pi r \cdot \frac{\theta}{2\pi} = \theta r & \text{and}\\ A(\theta) &= \pi r^2 \cdot \frac{\theta}{2\pi} = \frac{\theta}{2} r^2 \end{align*}