Angle units explained
Angles are measured in whatever someone once found convenient, and the choices stuck: 360 from Babylonian arithmetic, 400 from the metric reformers, and the radian from the circle itself.
Degree °
Base unit — every factor on this page is measured against it
One three-hundred-and-sixtieth of a full turn. The number survives because 360 divides so willingly — by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more — so halves, thirds and quarters of a circle all land on whole numbers.
Radian rad
1 rad = 57.29578 ° Exact
The circle's own unit: the angle where the arc along the circumference is as long as the radius. About 57.3 degrees, or exactly 180 divided by pi. Awkward as a number, and the only unit in which the calculus of sine and cosine comes out clean.
Milliradian mrad
1 mrad = 0.05729578 ° Exact
A thousandth of a radian. Used in optics for beam divergence and in shooting for scope adjustment, because at 100 metres one milliradian subtends almost exactly 10 centimetres — which makes range estimation arithmetic instead of guesswork.
Gradian gon
1 gon = 0.9 ° Exact
The metric attempt at the angle: 100 gradians in a right angle, 400 in a full circle. It never displaced the degree in general use but survives in European surveying, and most scientific calculators still offer a GRAD mode beside DEG and RAD.
Arcminute ′
1 ′ = 0.01666667 ° Exact
A sixtieth of a degree. Astronomy and navigation use it, and so does the eye: normal visual acuity is the ability to resolve detail one arcminute across, which is what the 20/20 line on a chart is testing.
Arcsecond ″
1 ″ = 0.000277777778 ° Exact
A sixtieth of an arcminute, or one 3,600th of a degree. The scale of telescope resolution and stellar parallax — the parsec is named for the distance at which a star shows one arcsecond of parallax over a year.
Turn turn
1 turn = 360 ° Exact
One full revolution, 360 degrees or two pi radians. The most intuitive unit and the least used in writing, though it is how rotation rates are actually counted: revolutions per minute is turns per minute.