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Angle Measurement Systems

A complete guide to geometric angle conversions: Degrees, Radians, Gradians, and military Mils.

Slicing the Circle

Angles measure rotation and geometry. There are primarily two ways humans have decided to slice a circle: based on arbitrary Babylonian math (360) or based on the mathematical constant $\pi$.

Babylonian Base 60

  • Degree (°): 1/360th of a full circle.
  • Arcminute ('): 1/60th of a degree. Crucial for astronomy and longitude/latitude.
  • Arcsecond (''): 1/60th of an arcminute.

Trigonometric Radians

  • Radian (rad): The angle made by taking the radius and wrapping it along the edge of the circle. Equivalent to $180/\pi$ degrees (approx 57.3°). It is the standard unit of angular measure in all higher mathematics and calculus.

Surveying and Military

  • Gradian / Gon: A metric attempt to decimalize angles, where a right angle is 100 grads (400 per circle). Used heavily in surveying.
  • NATO Mil: Used in artillery and sniper scopes. A circle is divided into 6400 equal mils.