Angle calculator

Every unit here divides the same full turn: 360 degrees, 2π radians, 400 gradians, 21,600 arcminutes, one turn. Only the divisor changes, so the conversions are all a single multiplication, and the reason there are five of them is that each was chosen to make a different job easy.

The mode switch is where the mistakes live

Trigonometric functions in software take radians and almost never say so. Type =SIN(30) into a spreadsheet expecting a half and you get −0.988032, because the function read 30 as radians and returned the sine of an angle nearly five full turns round. There is no error and no warning; the answer is a perfectly ordinary number in the right range. Spreadsheets provide RADIANS() and DEGREES() for the conversion, and most programming languages provide neither, leaving you to multiply by π/180 yourself.

A pocket calculator has the same trap wearing a different hat, in the DEG/RAD/GRAD indicator that nobody reads. If a trigonometric result is wrong by an amount that looks random rather than by a factor you recognise, the mode is the first thing to check and it is right far more often than the arithmetic.

Gradians, mils and the units built for a job

Gradians put a right angle at exactly 100 and a full turn at 400, so surveying arithmetic stays decimal from end to end. They survive on European surveying instruments and as a calculator key that is mostly pressed by accident. Arcminutes and arcseconds, at a sixtieth and a three-thousand-six-hundredth of a degree, remain standard in astronomy and navigation; the parsec is defined as the distance at which one astronomical unit subtends a single arcsecond, and normal visual acuity is roughly the ability to resolve one arcminute.

The milliradian is the unit that earns its keep by making a triangle disappear. One milliradian subtends one metre at a kilometre, near enough exactly, so range and size can be read off each other with no trigonometry at all. Be careful which mil you have: a true milliradian gives 6,283.19 to a full turn, while the NATO artillery mil rounds that to 6,400 for convenience. The NATO mil is therefore 0.05625 degrees and subtends 0.98 metres at a kilometre, about 2 per cent short of the rule of thumb.

What the type row is reading

Acute below 90 degrees, right at exactly 90, obtuse between 90 and 180, straight at exactly 180, reflex above. The classification is applied to the number as you entered it, without folding it into a single turn, so 400 degrees is reported as reflex rather than as the 40 degrees it points at. Wrap an accumulated rotation into the 0 to 360 range yourself before reading that row.