🔄 Degree (°) ↔ Radian (rad) Converter
Convert between degrees and radians.
Conversion Formula
One radian is the central angle subtended by an arc equal in length to the radius. Therefore, the circumference 2πr of a circle with radius r equals 2π radians.
Key Angle Conversion Table
| Degree (°) | Radian (rad) | π Expression |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 0.5236 | π/6 |
| 45° | 0.7854 | π/4 |
| 60° | 1.0472 | π/3 |
| 90° | 1.5708 | π/2 |
| 120° | 2.0944 | 2π/3 |
| 180° | 3.1416 | π |
| 270° | 4.7124 | 3π/2 |
| 360° | 6.2832 | 2π |
Unit Circle and Trigonometry
The point (cos θ, sin θ) on the unit circle (radius=1) visually represents angle θ. The red dot in the left diagram is this point.
Worked examples
180° is π ≈ 3.1415927 rad and 90° is π/2 ≈ 1.5707963 rad, the two conversions used most often.
One radian is about 57.2957795°, so 2 rad is roughly 114.5915590°.
What the conversion assumes
The conversion is exact up to rounding: degrees × π/180 is the definition, and π is irrational, so the decimal is always an approximation. Negative angles and values beyond 360° pass through unchanged — nothing is wrapped into a single turn. Radian output is shown in decimals and as a multiple of π when it is a clean fraction.
Frequently asked questions
- Why does a full turn equal 2π radians?
- A radian is the angle whose arc equals the radius, and a unit circle has circumference 2π, so a full turn spans 2π ≈ 6.2831853 rad.
- When should I use radians instead of degrees?
- In calculus and physics, because only radians make the derivative of sine equal to cosine without a factor; degrees stay convenient for geometry, navigation and everyday angles.
- Is there a quick mental conversion?
- Multiply degrees by about 0.01745 to get radians, or radians by 57.3 for degrees. So 30° is roughly 0.5236 rad, a handy sanity check.