An angle converter changes a rotation or bearing between units — degrees to radians, gradians to degrees, or arcseconds to degrees. Degrees are the everyday unit, radians are what mathematics and programming languages expect, and the smaller subdivisions appear in navigation, surveying and astronomy. This tool covers degrees, radians, gradians, arcminutes, arcseconds, turns and milliradians.
The formula. Every value passes through the degree. A full circle is 360 degrees = 2π radians = 400 gradians = 1 turn, which gives 1 radian = 180/π ≈ 57.29578° and 1 gradian = 0.9°. The subdivisions are sexagesimal: 1° = 60 arcminutes and 1 arcminute = 60 arcseconds, so one arcsecond is 1/3600 of a degree.
Worked example. A right angle is 90°, which is 90 × π/180 = 1.570796 radians (π/2) or 100 gradians. Half a circle, 180°, is exactly π ≈ 3.141593 radians. Going the other way, 1 radian is about 57.3° — a little under a sixth of a circle.
Where it's used. Programming is the most common reason people need this: JavaScript's Math.sin(), Python's math.cos() and almost every other trigonometric function take radians, not degrees, so a value from a design tool or a user input in degrees must be converted first. Beyond that, latitude and longitude use degrees, minutes and seconds; surveying instruments often read in gradians; and telescope pointing and optical tolerances are specified in arcseconds and milliradians.