Angle converter

How to convert an angle

1. Type the angle into either field. Degrees to radians and radians to degrees are the same control.
2. Pick the units — degrees, radians, gradians and turns in the first group, the sub-degree units used in astronomy, optics and ballistics in the second.
3. The formula panel shows the exact factor, and the chart below covers the standard angles in one go.

Degrees and radians

A full turn is 360 degrees or 2π radians, so the conversion is a factor of π/180, or about 0.01745329252. Going the other way, one radian is 180/π degrees, or 57.29577951°.

Radians are the natural unit for anything involving calculus or circular motion, because the radian is defined so that an arc of one radian on a unit circle has length one. That is what makes the derivative of sin(x) equal cos(x) — a result that quietly acquires a factor of π/180 if you insist on degrees. Every programming language's trigonometric functions expect radians, which is the most common practical reason to convert.

Arcminutes, arcseconds and coordinates

Degrees subdivide sexagesimally, exactly like hours: one degree is 60 arcminutes, one arcminute is 60 arcsecs. Latitude and longitude are still often written this way — 51° 28′ 38″ N — and converting to decimal degrees means dividing the minutes by 60 and the seconds by 3,600 before adding them on.

For scale, an arcsecond is roughly the angle subtended by a small coin at five kilometres, and one arcsecond of latitude is about 31 metres on the ground. Astronomers routinely work in milliarcseconds; the Gaia spacecraft measures stellar positions to a few tens of microarcseconds.

Milliradians, mils and shooting

A milliradian is a thousandth of a radian, and its appeal is the arithmetic: at 100 metres, one milliradian subtends almost exactly 10 centimetres, and at 1,000 metres exactly one metre. That makes range and holdover estimation a matter of moving a decimal point.

The NATO mil is a deliberate approximation of the same idea that divides the circle into 6,400 equal parts instead of 6,283.19. It gives 0.05625° per mil against a true milliradian's 0.0572958°, a difference of about 2 %. Artillery uses the 6,400 version; most rifle scopes use true milliradians. Both are in the unit list, separately labelled, because mixing them at distance puts rounds well off target.

Gradians, and why your calculator has a DRG key

The gradian divides a right angle into 100 parts, so a full turn is 400 gon. It came out of the same French revolutionary push for decimalisation that gave us the metre, and survives in European surveying and in the "GRAD" mode on scientific calculators. If your trigonometry answers are inexplicably wrong by a consistent factor, checking whether the calculator is in DEG, RAD or GRAD mode is the first thing to try.