Convert degrees to radians: formula, table and examples
Convert angles from degrees to radians and back. Formula, common angles table and trigonometry applications.
Published on January 15, 2026Converting 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.
Understanding the conversion
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
📊 Conversion 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
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.
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.
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.