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
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