This free angle converter changes radians (rad) to degrees (°) and back again, useful for students, engineers, and anyone working with trigonometry, robotics, or graphics where one unit system is required but a value is given in the other. Type a value in either field and the result updates instantly.
How it works
An angle measured in radians is the arc length it subtends on a circle of radius 1. A full turn around that circle has circumference 2π, and a full turn is 360°, so 2π radians = 360°, which simplifies to π radians = 180°. Solving for degrees gives the exact conversion formula:
degrees = radians × 180 ÷ π
To go the other way, radians = degrees × π ÷ 180. The constant 180 ÷ π ≈ 57.29578 is the number of degrees in a single radian. The tool applies these formulas directly with full floating-point precision and rounds the displayed result for readability.
Example
Convert 2.5 radians to degrees:
degrees = 2.5 × 180 ÷ π = 450 ÷ 3.14159265 ≈ 143.2394°
Going back: 143.2394 × π ÷ 180 = 143.2394 × 0.0174533 ≈ 2.5 radians, confirming the round trip.
Common angle reference values
These are the angles most frequently encountered in trigonometry, and knowing them in both systems is essential for switching between contexts without reaching for a calculator every time:
| Radians (rad) | Degrees (°) | Notes |
|---|---|---|
| 0 | 0° | Origin |
| π/6 ≈ 0.5236 | 30° | sin = 0.5 |
| π/4 ≈ 0.7854 | 45° | sin = cos = √2/2 |
| π/3 ≈ 1.0472 | 60° | cos = 0.5 |
| π/2 ≈ 1.5708 | 90° | Quarter turn |
| π ≈ 3.1416 | 180° | Half turn |
| 3π/2 ≈ 4.7124 | 270° | Three-quarter turn |
| 2π ≈ 6.2832 | 360° | Full turn |
Why radians are used in mathematics and engineering
Degrees are intuitive for describing everyday angles, but radians are the natural unit for calculus and most physics because they make the math simpler. The derivative of sin(x) is cos(x) — but only when x is in radians. If you differentiate sin(x°) with respect to x, you get (π/180)cos(x°), which is an unwanted constant everywhere.
Similarly, the arc length formula is simply arc = rθ when θ is in radians, compared to arc = rθπ/180 in degrees. Taylor series expansions for sin and cos also assume radian input.
Programming languages and graphics libraries almost universally use radians. JavaScript’s Math.sin(), Python’s math.sin(), and OpenGL shader functions all take radians. Entering a degree value directly will give a wrong result — convert first, then pass the radian value.
Negative angles and angles over 360°
The converter handles negative angles (measuring clockwise instead of counter-clockwise) and angles greater than 2π/360° without any issue. For example, 720° is 4π radians — two full rotations — and −90° is −π/2 radians. In most contexts, angles are reduced modulo 2π (or 360°) when comparing positions, but the conversion itself is linear and works for any value.
All calculations run entirely in your browser — nothing you type is uploaded or stored.