Radians and π

One radian is the measure of a central angle whose intercepted arc has the same length as its radius. If the arc length is twice the radius, the central angle has a measure of 2 radians, etc. You see below that as the point swings around the circle, the central angle and the intercepted arc both increase. Since the arc length (s) is just a part of the circumference, we can express it as a fractional part of the circumference. The central angle is θ radians, out of a possible 2π radians.

So, s = (θ/(2π))×(2πr)  = θr.

Divide both sides by r to get the definition of radian measure; it is the ratio of the arclength to the radius. θ = s/r