AHL 3.7—Radians

Unlike degrees — which divide a circle into 360 arbitrary parts — the radian is defined geometrically from the circle itself. This makes it the most natural unit for angles in advanced mathematics.

Imagine "unrolling" the radius of a circle along its circumference. The angle swept out at the centre when you've laid exactly one radius-length along the arc is 1 radian.

Since the full circumference of a circle is $2\pi r$, fitting the radius around the circle $2\pi$ times gives us the key relationship:

$360°=2\pi \text{ radians}$

and therefore:

$180°=\pi \text{ radians}$