Angle Units Guide — Degrees, Radians & Gradians
Angles are measured in several different units depending on the context. Mathematics uses radians, everyday life uses degrees, and surveying uses gradians. Here's a clear guide to all of them.
Degrees (°)
The most familiar angle unit. A full circle = 360°. A right angle = 90°. A straight line = 180°. Used in navigation, geometry, maps, and everyday contexts.
Radians (rad)
The SI unit of angle. 1 radian is the angle where the arc length equals the radius. A full circle = 2π radians ≈ 6.2832 rad. π radians = 180°. Used in calculus, physics, and engineering because it simplifies formulas.
Gradians (grad)
Also called gon or grade. A full circle = 400 gradians, so a right angle = 100 gradians. Used mainly in surveying, civil engineering, and some European countries.
Arcminutes and Arcseconds
Subdivisions of a degree. 1° = 60 arcminutes (') = 3600 arcseconds (''). Used in astronomy (star positions), GPS coordinates, and precision optics.
Conversion Table
| Degrees | Radians | Gradians | Arcminutes |
|---|---|---|---|
| 0° | 0 | 0 | 0' |
| 30° | π/6 ≈ 0.5236 | 33.33 | 1,800' |
| 45° | π/4 ≈ 0.7854 | 50 | 2,700' |
| 90° | π/2 ≈ 1.5708 | 100 | 5,400' |
| 180° | π ≈ 3.1416 | 200 | 10,800' |
| 360° | 2π ≈ 6.2832 | 400 | 21,600' |
📐 Key fact: In calculus and physics, always use radians — not degrees. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, you get an extra π/180 factor.