Radians

A circle's own unit of angle

Degrees are a human invention — someone chose to cut a full turn into 360 equal slices. A radian is the angle the circle chooses for itself: it is the angle that wraps an arc equal in length to the radius around the circle.

Walk the radius-length around the rim and the angle you sweep is exactly one radian (about 57.3^\circ). Since the whole circumference is 2\pi r — that's 2\pi radius-lengths — a full turn is

360^\circ = 2\pi \text{ radians}, \qquad \text{so } 180^\circ = \pi \text{ radians}.

That single fact is the whole conversion. To go degrees → radians, multiply by \dfrac{\pi}{180}; to go radians → degrees, multiply by \dfrac{180}{\pi}.

The values worth knowing

From 180^\circ = \pi the common angles fall out by simple division, and the radian makes the two circle formulas as clean as they can be.

a full turn 2\pi \text{ rad} = 360^\circ, and a half turn \pi \text{ rad} = 180^\circ;
common values: 30^\circ = \tfrac{\pi}{6}, 45^\circ = \tfrac{\pi}{4}, 60^\circ = \tfrac{\pi}{3}, 90^\circ = \tfrac{\pi}{2};
arc length s = r\theta and sector area A = \tfrac{1}{2}r^2\theta — but only when \theta is measured in radians.

See one radian

Step through the figure: lay down the radius, wrap an arc of the same length to sweep one radian, then continue to the half-circle, which is \pi radians.