Converting between degrees and radians is essential in mathematics, physics and programming. Trigonometric functions in most programming languages (sin, cos, tan in JavaScript, Python, C++) use radians, not degrees. Understanding this conversion prevents many common bugs.
The radian is the SI unit for angles. A full circle equals 2π radians = 360°. The conversion is proportional: to convert degrees to radians, multiply by π/180 (≈ 0.01745). To convert radians to degrees, multiply by 180/π (≈ 57.296). Using radians simplifies many mathematical formulas (arc length, derivatives of trig functions).
📐 Formula
Radians = Degrees × π / 180 | Degrees = Radians × 180 / π
📊 Reference table
| Degrees | Radians (exact) | Radians (decimal) | Position |
|---|---|---|---|
| 0° | 0 | 0 | East (right) |
| 30° | π/6 | 0.5236 | — |
| 45° | π/4 | 0.7854 | Diagonal |
| 60° | π/3 | 1.0472 | — |
| 90° | π/2 | 1.5708 | North (up) |
| 180° | π | 3.1416 | West (left) |
| 270° | 3π/2 | 4.7124 | South (down) |
| 360° | 2π | 6.2832 | Full circle |
💡 Practical examples
Example 1: using sin() in JavaScript
Math.sin(90) gives ~0.894, not 1! You must convert: Math.sin(90 * Math.PI / 180) = Math.sin(π/2) = 1. Always convert degrees to radians before calling a trig function.
Example 2: arc length of a circle
Circle of radius 5, arc of 60°: angle in radians = 60 × π/180 = π/3. Arc length = r × θ = 5 × π/3 ≈ 5.24 units.
Example 3: slope angle
A ramp at 15° = 15 × π/180 = 0.2618 rad. Tangent of 15° = tan(0.2618) ≈ 0.268. For a 10m ramp: height = 10 × 0.268 = 2.68m.