Introduction to Degrees and Radians

Angles can be measured in degrees or in radians; degrees are the everyday unit for 'how big a turn', while radians tie an angle directly to the geometry of a circle and are the natural unit throughout circular motion, calculus, and most of physics.

A radian is defined as the angle subtended at the centre of a circle by an arc whose length exactly equals the circle's radius. In general, for an arc of length $s$ on a circle of radius $r$, the angle it subtends (in radians) is

Going all the way around a full circle, the arc length is the circumference, $2\pi r$, so the full circle subtends an angle of $2\pi r/r = 2\pi$ radians — and a full circle is also, by the everyday degree convention, $360°$. Equating the two gives the conversion:

$360° = 2\pi \text{ rad} \qquad\Longrightarrow\qquad 1 \text{ rad} = \frac{180°}{\pi} \approx 57.295°$ …