Angles measure the rotation between two rays from a common point. Degrees divide a full circle into 360 equal parts — a number the Babylonians chose for its exceptional divisibility (divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180). Radians are the natural mathematical unit: a radian is defined as the angle whose arc length equals the radius, which simplifies calculus and trigonometric identities significantly.
The gradian (grad) divides a right angle into 100 equal parts (full circle = 400 grad), introduced alongside the metric system to make surveying arithmetic easier. Arcminutes (1/60°) and arcseconds (1/3600°) are used in astronomy, navigation, and cartography — the Moon's diameter spans about 30 arcminutes as seen from Earth. Milliradians (mrad) are used in optics and ballistics: at a distance of 1 km, 1 mrad corresponds to exactly 1 m, making range estimation straightforward.
| Full circle | = 360° = 2π rad = 400 grad |
| 1 rad | = 57.2958° |
| 1° | = π/180 rad |
| 1° | = 60 arcmin = 3,600 arcsec |
| 1 mrad | = 0.05730° |
| 1 turn | = 360° |