Angle Converter

Radians are the mathematical unit

One radian is the angle subtending an arc equal to the radius, which makes a full turn exactly 2π radians. That definition is why calculus works cleanly in radians and awkwardly in degrees: the derivative of sine is cosine only when the argument is in radians. Programming language trigonometric functions almost universally expect radians, and passing degrees is one of the most common numerical bugs there is, producing plausible-looking but entirely wrong output rather than an error.

The conversions

Degrees to radians is multiply by π and divide by 180. Radians to degrees is the reverse. Useful anchors: 180 degrees is π radians, 90 is π/2, 60 is π/3, 45 is π/4 and 30 is π/6. Those five cover most of what appears in practice.

Gradians exist and are rarely wanted

A gradian divides a right angle into 100 parts, so a full turn is 400 gradians. It was introduced with the metric system to decimalise angle measurement and never displaced degrees except in some surveying. Calculators still offer a GRAD mode, and a calculator accidentally left in it is a classic source of trigonometry answers that are close enough to look right and wrong enough to matter.

Mils in artillery and optics

The milliradian is one thousandth of a radian, and its usefulness is a small-angle approximation: at 1,000 metres, one milliradian subtends almost exactly one metre. That relationship makes range and adjustment arithmetic trivial in the field. Confusingly, NATO uses 6,400 mils per turn rather than the mathematically exact 6,283, rounded for divisibility, so mil is not always the same unit.

Bearings run the other way

Navigation bearings measure clockwise from north; mathematical angles measure anticlockwise from east. Converting between them requires both a rotation and a reflection, not just an offset, which is why naively subtracting 90 degrees gives an answer that is right in one quadrant and wrong in the others.