The formula
radians = degrees × π ÷ 180
Where it comes from
A radian is the angle you get when the arc equals the radius, which makes it the natural unit for anything involving circles. Degrees are a historical convention from Babylonian astronomy: 360 divides cleanly by a great many numbers, which is handy and arbitrary.
How to work it out by hand
- A full turn is 360°, 2π radians, 400 gradians or 1 turn
- To go from degrees to radians, multiply by π and divide by 180
- To go back, multiply by 180 and divide by π
- Express the result as a multiple of π to keep it exact
What is worth knowing
Radians are not a convenience, they are a requirement: the derivative of sin x is cos x only when x is in radians, and every series expansion in analysis assumes them. In degrees the derivative picks up a factor of π/180, which is why calculus is never done in degrees. The other thing worth knowing is that a radian is not a unit in the dimensional sense — it is a ratio of two lengths, so it cancels out — which is why you can write sin(2) with no unit at all and be understood.