Common angle conversions

Degrees	Radians	Gradians	Turns
0°	0 rad	0 grad	0 turn
30°	π/6 ≈ 0.5236 rad	33.33 grad	1/12 turn
45°	π/4 ≈ 0.7854 rad	50 grad	1/8 turn
90°	π/2 ≈ 1.5708 rad	100 grad	0.25 turn
180°	π ≈ 3.1416 rad	200 grad	0.5 turn
360°	2π ≈ 6.2832 rad	400 grad	1 turn

How angle conversion works

All conversions pass through degrees as the intermediate unit. Multiply the input by its "to degrees" factor, then divide by the target unit's factor.

The conversion formula between degrees and radians is: radians = degrees × π / 180.

Where each unit is used

Degrees (°): Navigation, carpentry, everyday geometry, and most consumer applications. Familiar to almost everyone.
Radians (rad): Mathematics, physics, computer graphics APIs (CSS transforms, WebGL, canvas). The natural unit for calculus.
Gradians (grad): Surveying, military artillery, and civil engineering - especially common in continental Europe. A full circle = 400 gradians, making right angles a clean 100 grad.
Turns: Clock design, robotics, and full-rotation contexts. One turn = 360° = 2π rad = 400 grad.

Why radians are the natural unit in mathematics

A radian is defined so that an arc of length r on a circle of radius r subtends exactly 1 radian at the center. This makes arc length = r × θ (with θ in radians), a clean, dimensionless relationship with no conversion factor. Consequently, the derivatives of trigonometric functions are simple: d/dx[sin x] = cos x works only when x is in radians. Using degrees introduces a factor of π/180 everywhere in calculus.

Common angles in context

Degrees	Radians	Context / meaning
30°	π/6	Shorter leg of a 30–60–90 right triangle
45°	π/4	Diagonal of a square; isoceles right triangle
60°	π/3	Interior angle of an equilateral triangle
90°	π/2	Right angle; perpendicular lines
120°	2π/3	Interior angle of a regular hexagon
180°	π	Straight line; supplementary angles sum to this
270°	3π/2	Three-quarter rotation
360°	2π	Full circle; complete rotation