Degrees ↔ Radians Converter
Type in either box to convert instantly.
Common Angles Reference Table
| Degrees | Exact Radians | Decimal Radians | sin | cos |
|---|---|---|---|---|
| 0° | 0 | 0.0000 | 0 | 1 |
| 30° | π/6 | 0.5236 | 0.5 | 0.8660 |
| 45° | π/4 | 0.7854 | 0.7071 | 0.7071 |
| 60° | π/3 | 1.0472 | 0.8660 | 0.5 |
| 90° | π/2 | 1.5708 | 1 | 0 |
| 120° | 2π/3 | 2.0944 | 0.8660 | −0.5 |
| 135° | 3π/4 | 2.3562 | 0.7071 | −0.7071 |
| 150° | 5π/6 | 2.6180 | 0.5 | −0.8660 |
| 180° | π | 3.1416 | 0 | −1 |
| 270° | 3π/2 | 4.7124 | −1 | 0 |
| 360° | 2π | 6.2832 | 0 | 1 |
Degrees to Radians – The Formula
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.
- Degrees → Radians: radians = degrees × (π ÷ 180)
- Radians → Degrees: degrees = radians × (180 ÷ π)
Why Radians?
Radians are the natural unit of angular measurement in mathematics. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. This definition makes calculus, trigonometry identities, and Fourier analysis much cleaner — for example, the derivative of sin(x) with respect to x equals cos(x) only when x is measured in radians.
Frequently Asked Questions
Why is a full circle 2π radians?
The circumference of a unit circle (radius = 1) is 2π × 1 = 2π. Since one radian equals one radius-length of arc, a full circle corresponds to 2π radians — the ratio of circumference to radius.
When should I use degrees vs radians?
Use degrees for navigation, surveying, everyday geometry, and anything involving angles people will read. Use radians in calculus, physics equations, programming (most math library functions like Math.sin() expect radians), and engineering calculations.
What is π exactly?
π (pi) is an irrational number approximately equal to 3.14159265358979. It cannot be expressed as a simple fraction and its decimal expansion never repeats. It represents the ratio of a circle's circumference to its diameter.