Degrees to Radians Converter
This angle converter transforms degrees (°) into radians (rad) and back again in real time. Degrees divide a full rotation into 360 equal parts — a convention inherited from Babylonian astronomy. Radians measure angles in terms of arc length on a unit circle, which makes them the natural unit for all higher mathematics and nearly every programming math library. Converting fluently between the two is an everyday task for students, engineers, game developers, and data scientists.
Why radians exist and when you need them
The radian is defined so that an angle of 1 rad subtends an arc equal in length to the circle’s radius. A full circle has a circumference of 2πr, so a full rotation is 2π radians. This definition makes trigonometric identities, derivatives, and Taylor series clean:
- The derivative of sin(x) is cos(x) only when x is in radians. In degrees it becomes (π/180)·cos(x).
Math.sin(),Math.cos(), andMath.tan()in JavaScript, Python, C, Java, and virtually every other language expect radians by default.- Physics formulas for angular velocity (ω = θ/t), centripetal acceleration, and wave equations all use radians.
- Game engines and 3D rotation matrices use radians internally even when the UI shows degrees.
How the conversion works
The fundamental relationship follows from a full circle equalling 360° or 2π rad:
radians = degrees × π ÷ 180
degrees = radians × 180 ÷ π
Type a value in either field and the converter applies the corresponding formula instantly. Because π is irrational its decimal expansion never terminates, so radian values are displayed to several significant figures; common angles like 30°, 45°, 60°, 90°, and 180° map to exact fractions of π.
Complete reference table
| Degrees (°) | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 150° | 5π/6 | 2.617994 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
Common conversion errors to avoid
- Passing degrees to
Math.sin()without converting is among the most frequent bugs in trigonometry-based code. The result looks plausible for small angles (sin(5°) ≈ 0.087 but sin(5 rad) ≈ −0.959) making it hard to spot. - Dividing by 180 without multiplying by π — forgetting the π factor off-by-a-factor bug.
- Forgetting that negative angles are valid (−90° = −π/2 rad) and that the converter works in both directions for any real number.
All calculations run entirely in your browser — no data is sent to any server.