Radian measure and arc length

In a circle of radius 1 unit (a unit circle), the angle at the centre made by an arc 1 unit long measures 1 radian. The same holds in any circle: an arc as long as the radius makes 1 radian at the centre.

The whole circumference of a unit circle is 2π, so one complete revolution is 2π radians, a half turn is π radians and a quarter turn is π/2 radians.

An arc twice as long makes twice the angle at the centre, so the angle is proportional to the arc. An arc of length l in a circle of radius r therefore makes θ = l/r radians at the centre; equivalently l = rθ.

l = rθ holds only with θ in radians. For example, with π = 22/7, an arc of a circle of radius 21 cm making 60° = π/3 at the centre has length 21 × π/3 = 7π = 22 cm.

A radian is a ratio of two lengths, so it has no dimensions; the word radian is often left out, and an angle written without a degree sign is read in radians.

Radians match the real numbers: lay a number line along the tangent at a point A of the unit circle with 0 at A, and wrap the positive half anticlockwise and the negative half clockwise round the circle. Each real number x then lands at the point reached by an angle of x radians.