Degrees to Radians Converter (° to rad)
Convert degrees to radians (° to rad). See exact π fractions, decimal radians, and sin/cos values. Essential for trigonometry, calculus, programming, and engineering.
ANGLE IN RADIANS
0.785398 rad
EXACT
π/4
TURNS
0.1250
SIN(45°)
0.707107
COS(45°)
0.707107
Degrees to Radians — Key Angles
| Degrees | Radians (exact) | Radians (decimal) | sin | cos |
|---|---|---|---|---|
| 0° | 0 | 0.0000 | 0.0000 | 1.0000 |
| 30° | π/6 | 0.5236 | 0.5000 | 0.8660 |
| 45° | π/4 | 0.7854 | 0.7071 | 0.7071 |
| 60° | π/3 | 1.0472 | 0.8660 | 0.5000 |
| 90° | π/2 | 1.5708 | 1.0000 | 0.0000 |
| 120° | 2π/3 | 2.0944 | 0.8660 | -0.5000 |
| 135° | 3π/4 | 2.3562 | 0.7071 | -0.7071 |
| 150° | 5π/6 | 2.6180 | 0.5000 | -0.8660 |
| 180° | π | 3.1416 | 0.0000 | -1.0000 |
| 270° | 3π/2 | 4.7124 | -1.0000 | -0.0000 |
| 360° | 2π | 6.2832 | -0.0000 | 1.0000 |
| 720° | 4π | 12.5664 | -0.0000 | 1.0000 |
How to Convert Degrees to Radians
Multiply by π/180 (approximately 0.01745):
Example: 45° (diagonal)
= 45 × π/180 = π/4 ≈ 0.7854 rad
Example: 90° (right angle)
= 90 × π/180 = π/2 ≈ 1.5708 rad
Example: 180° (straight line)
= 180 × π/180 = π ≈ 3.1416 rad
Why π/180? A full circle = 360° = 2π radians. So 1° = 2π/360 = π/180 radians. The radian is the "natural" angle unit because an arc of length equal to the radius subtends exactly 1 radian.
Key Angles — Degrees to Radians
| Degrees | Radians (exact) | Radians (decimal) | sin | cos | tan |
|---|---|---|---|---|---|
| 0° | 0 | 0.0000 | 0 | 1 | 0 |
| 30° | π/6 | 0.5236 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | 0.7854 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1.0472 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1.5708 | 1 | 0 | ∞ |
| 120° | 2π/3 | 2.0944 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | 2.3562 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 2.6180 | 1/2 | −√3/2 | −√3/3 |
| 180° | π | 3.1416 | 0 | −1 | 0 |
| 270° | 3π/2 | 4.7124 | −1 | 0 | ∞ |
| 360° | 2π | 6.2832 | 0 | 1 | 0 |
Radians in Programming Languages
| Language | Trig Functions Use | Convert Degrees → Radians |
|---|---|---|
| JavaScript | Radians | Math.sin(degrees * Math.PI / 180) |
| Python | Radians | math.radians(degrees) or numpy.deg2rad() |
| C / C++ | Radians | sin(degrees * M_PI / 180.0) |
| Java | Radians | Math.toRadians(degrees) |
| C# | Radians | degrees * Math.PI / 180 |
| CSS | Both | transform: rotate(1.5708rad) or rotate(90deg) |
Every major programming language uses radians for trigonometric functions. If you pass degrees to Math.sin() or sin(), you'll get wrong results. Always convert first!
Unit Circle — Quadrant Reference
| Quadrant | Degrees | Radians | sin | cos |
|---|---|---|---|---|
| I | 0°–90° | 0–π/2 | + | + |
| II | 90°–180° | π/2–π | + | − |
| III | 180°–270° | π–3π/2 | − | − |
| IV | 270°–360° | 3π/2–2π | − | + |
Angle Measurement Systems
| System | Full Circle | Right Angle | Used By |
|---|---|---|---|
| Degrees (°) | 360° | 90° | Navigation, construction, everyday |
| Radians (rad) | 2π | π/2 | Math, physics, programming |
| Gradians (gon) | 400 gon | 100 gon | European surveying |
| Turns | 1 turn | 0.25 turns | Engineering, astronomy |
What Is a Degree (°)?
A degree is 1/360th of a full rotation. The 360-degree system dates back to ancient Babylonians (who used base-60 math). Degrees are used in everyday life for navigation, construction, weather (wind direction), and education. In the US, degrees are taught first in geometry class before students encounter radians in precalculus or AP Calculus.
What Is a Radian?
A radian is the angle subtended by an arc equal in length to the radius of the circle. One full circle = 2π radians ≈ 6.2832 rad. Radians are the SI unit for angles and are required by virtually all programming languages and scientific calculations. They make calculus formulas cleaner: d/dx sin(x) = cos(x) only works when x is in radians.