Angle Converter

Why radians are the “natural” unit

One radian is the angle that sweeps an arc equal in length to the radius. That makes arc length simply rθ with no conversion constant, and it is the reason calculus formulas for sine and cosine only work cleanly in radians. The derivative of sin(x) is cos(x) only when x is in radians; in degrees you pick up a factor of π/180. A full turn is 2π radians, which is why π keeps appearing in places that seem to have nothing to do with circles.

Degrees, minutes and seconds

The 360-degree circle comes from Babylonian base-60 arithmetic, and the subdivisions kept that base: one degree is 60 arcminutes, one arcminute is 60 arcseconds. Latitude and longitude are still written this way — 40°26′46″ N — because one arcminute of latitude is one nautical mile, which makes chart work direct. An arcsecond of latitude is about 31 metres, which is roughly the precision a coordinate written to whole seconds implies.

Gradians and mils: the units nobody agrees on

A gradian divides the circle into 400, so a right angle is exactly 100 gon. It was a metric-era attempt at decimalising angles and survives mainly in European surveying. The NATO mil divides the circle into 6400, chosen so that one mil subtends roughly one metre at a kilometre — handy for artillery. A true milliradian subtends exactly one metre at a kilometre, but the circle contains 6283.19 of them, which is awkward, so several incompatible “mil” conventions exist: 6400 (NATO), 6000 (Warsaw Pact) and 6300 (Sweden). Check which one an instrument uses before trusting it.

Frequently Asked Questions