Degrees ⇄ radians

What it's for

Degrees are used to say how wide an angle is; radians are used when that angle goes into a formula. They're the same thing measured with two different rulers, and moving from one to the other is needed in trigonometry, in physics, in technical drawing and every time you program something that turns.

The conversion, and the number to remember

Half a turn is 180° = π radians: everything comes from there. A full turn is 360° = 2π; a right angle is 90° = π/2 ≈ 1.5708; 60° is π/3 ≈ 1.0472; 45° is π/4 ≈ 0.7854; 30° is π/6 ≈ 0.5236. For mental arithmetic: one radian is a little under 57.3 degrees, a fairly wide angle, more than half a right angle.

Why radians exist

Because the degree is a convention (the 360 degrees come from the Babylonians, not from geometry) while the radian is a natural measure: an angle of one radian is the one that cuts out on the circle an arc as long as the radius. The convenience follows from that definition: the length of an arc is simply radius × angle, with no conversion factor, and in physics rotation speed is written the same way.

The mistake everyone makes the first time

In programming languages the sine, cosine and tangent functions want radians, not degrees, and they don't say so: passing 90 instead of 1.5708 doesn't raise an error, it produces a wrong number that looks plausible. It applies to JavaScript, Python, C, Java and to spreadsheets. If a drawing turns a little when it should turn a quarter, that's nearly always the cause.

There's a third unit that shows up on scientific calculators too, the gradian (400 to a turn, the GRAD key): if your trigonometry results are all slightly odd, check the calculator isn't on that instead of DEG.

Nearby tools

For the other angle units, including degrees with minutes and seconds as used in navigation, there is Angle converter; for geographic coordinates GPS coordinates. For sums with sines and cosines the page is Scientific calculator, and for triangles Solve a triangle.