Bidirectional Degrees & Radians Converter | Angle Tool
Seamlessly convert angles between degrees and radians. Get instant decimal updates, exact fractional π values, and visualize quadrants in real-time.
Type any degree value.
Type a radian value (e.g., 3.14).
Summary
Calculation Steps
Unit Circle Visualization
This calculator is free, fast and ad-free. No sign-up, and what you type in isn't sent to us.
How it Works & Assumptions
This tool uses standard trigonometric formulas to convert your input instantly. We assume all angles are in standard position (starting from the positive x-axis and rotating counter-clockwise). This ensures we can accurately determine the quadrant (I, II, III, or IV) for a given angle.
The Core Formulas
Converting between these two units is straightforward once you understand the relationship: 180° equals π radians.
- To Radians: Multiply your Degree value by
π / 180. - To Degrees: Multiply your Radian value by
180 / π. - Exact Fractions: To get the "math class" answer (like 3π/2), we do not multiply by the approximate value of Pi (≈ 3.1416). Instead, we treat Pi as a symbol, find the fraction of the degrees over 180, and simplify it by finding the Greatest Common Divisor (GCD).
Quick Conversion Reference
| Degrees (°) | Radians (Exact) | Radians (Decimal) |
|---|---|---|
| 0° | 0 | 0.0000 |
| 30° | π/6 | ≈ 0.5236 |
| 45° | π/4 | ≈ 0.7854 |
| 60° | π/3 | ≈ 1.0472 |
| 90° | π/2 | ≈ 1.5708 |
| 180° | π | ≈ 3.1416 |
| 270° | 3π/2 | ≈ 4.7124 |
| 360° | 2π | ≈ 6.2832 |
Why Convert Between Degrees and Radians?
- Programming & Game Dev: Most math libraries (like JavaScript's
Math.sin()) exclusively accept radians. If you feed them degrees, your game physics will break. - Engineering: Torque, angular velocity, and rotational motion calculations rely on radians to keep formulas clean (avoiding the constant need for π/180 factors).
- Astronomy: Precise measurements of angular distance in the sky often require conversion between these units.
Frequently Asked Questions
How do you simplify degrees to radians with Pi?
Divide your degrees by 180. If you have 90°, 90/180 becomes 1/2. Then just put Pi on top: π/2. This gives you the exact radian value without messy decimals.
Is 1 radian equal to 180 degrees?
No, π (≈ 3.1416) radians is equal to 180 degrees. 1 radian is approximately 57.2958 degrees.