Angle converters

4 conversions between 6 angle units. Every page includes a live converter, the formula and a full conversion table.

Units covered

All angle conversions

• Degrees to Radians
• Degrees to Gradians
• Degrees to Turns
• Radians to Degrees

Why 360 degrees

The division is Babylonian and survives because 360 has an unusual number of divisors — 24 of them — so halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths of a circle are all whole numbers of degrees. Nothing about the choice is natural; it is simply convenient, and no proposed replacement has ever displaced it.

Radians are the natural unit

A radian is the angle subtended by an arc equal in length to the radius, which makes a full turn 2π radians. It looks awkward and is in fact the unit that makes the mathematics simple: the derivative of sin x is cos x only when x is in radians, and arc length is just radius times angle. Every trigonometric function in a programming language expects radians, which is the single most common source of wrong answers in graphics and geometry code — Math.sin(90) is not 1.

Minutes, seconds and gradians

A degree divides into 60 arcminutes and each of those into 60 arcseconds, the same sexagesimal scheme as clock time and from the same source. Latitude and longitude are still written this way, and one arcminute of latitude is one nautical mile by definition. The gradian divides the circle into 400 instead, so a right angle is a round 100 — a metric-era proposal that survives in some European surveying and on scientific calculators, and almost nowhere else.