Angle converter

This converter treats the degree as its base unit, so every factor describes how many degrees one unit of the other kind equals. A radian equals 57.29577951 degrees, a gradian equals 0.9 degrees, and a minute of arc — the kind used in old-fashioned navigation and telescope eyepieces — equals 1/60 of a degree, with the second of arc one sixtieth smaller again.

The pair people look up most is degree to radian. Divide the full-circle count of degrees (360) by 2π to get the degree-to-radian multiplier, or simply use: rad = ° × 0.01745329252. Reversing it means multiplying by 57.29577951 instead.

A right angle of 90° converts to 90 × 0.01745329252 = 1.570796 rad, which is exactly π/2 — a handy sanity check that the formula is working. A 45° angle converts to 45 × 1.111111 = 50 gradians, since a gradian is defined so that a full right angle equals exactly 100 of them.

Degrees dominate everyday geometry, GPS coordinates and compass headings, but other units survive in specific trades. Surveyors and some European engineering disciplines use gradians (also called gons) because 100 gradians make a right angle, which simplifies decimal calculations. Radians are the default in calculus, physics and programming, since trigonometric derivatives only come out clean when angles are measured in radians. Astronomers and riflescope manufacturers rely on arc-minutes and arc-seconds to describe tiny angular separations — a telescope's resolving power or a scope's windage adjustment is usually specified in minutes of arc.

Military angle units are a frequent source of error because "mil" is not standardized the same way everywhere. This converter uses the NATO mil, defined as 1/6400 of a circle (0.05625°), but some other systems have historically divided the circle into 6000 units instead. Mixing up the two changes an artillery bearing or compass adjustment by several percent, so always confirm which mil convention a manual or instrument is using before converting.

The remaining units describe fractions of a full turn rather than small angular slivers. A quadrant is a quarter turn (90°), a sextant a sixth of a turn (60°), and a sign — borrowed from astrology and old astronomy — a twelfth of a turn (30°), matching the twelve zodiac divisions of the sky. The 32-point compass rose used in sailing splits the circle into points of 11.25° each, which is why old nautical directions like "north-northeast" correspond to clean multiples of that value.