Degrees split a circle into 360 parts (a Babylonian choice). Radians split it into 2π parts — and 2π is no accident: it comes from the actual geometry of a circle.
1 radian
= 180°/π ≈ 57.296°
The angle made by an arc whose length equals the radius.
Common conversions
- 360° = 2π rad (one full turn)
- 180° = π rad (half turn / straight line)
- 90° = π/2 rad (right angle)
- 60° = π/3 rad
- 45° = π/4 rad
- 30° = π/6 rad
Calculus prefers radians because d/dx(sin x) = cos x ONLY when x is in radians. In degrees you'd pick up an awkward π/180 factor every time you differentiate.