Convert radians to degrees
PrintRadian to Degree Conversion Table
| Radian (rad) | Degree (°) |
|---|---|
| 0.01 rad | 0.5729578° |
| 0.1 rad | 5.729578° |
| 0.5 rad | 28.64789° |
| 1 rad | 57.29578° |
| 1.570796 rad (\(\pi / 2\)) | 90.0° |
| 3.141593 rad (\(\pi\)) | 180.0° |
| 6.283185 rad (\(2\pi\)) | 360.0° |
| 10 rad | 572.9578° |
In geometry, trigonometry, and calculus, angular measurements are defined in either degrees (°) or radians (rad). While degrees divide a circular rotation into 360 units based on ancient astronomical tracking, radians represent the ratio of arc length to radius ($s = r\theta$), making radians the standard unit for all mathematical derivations.
Our free online Degrees to Radians Converter provides instant angle conversions in both formats:
- Degrees to Exact π Radians: Reduces the angle fraction
Degrees / 180to simplest terms withπ. - Degrees to Decimal Radians: Multiply degrees by
π / 180(approx.0.0174532925). - Radians to Degrees: Multiply radians by
180 / π(approx.57.29577951).
Degrees to Radians Formula & Step-by-Step Method
To convert an angle from degrees to radians, multiply the degree value by the conversion factor π / 180:
Angle in Radians (rad) = Angle in Degrees (°) × (π / 180)
Step-by-Step Fraction Simplification Method
To express degrees as an exact fraction of π:
- Set up the ratio:
(Degrees / 180) × π - Find the greatest common divisor (GCD) of the degree value and 180.
- Divide both numerator and denominator by the GCD to simplify the fraction.
| Degree Angle (°) | Initial Ratio | GCD Divisor | Exact π Radians | Decimal Radians |
|---|---|---|---|---|
| 30° | 30π / 180 |
Divide by 30 | π / 6 rad | 0.5236 rad |
| 45° | 45π / 180 |
Divide by 45 | π / 4 rad | 0.7854 rad |
| 60° | 60π / 180 |
Divide by 60 | π / 3 rad | 1.0472 rad |
| 90° (Right Angle) | 90π / 180 |
Divide by 90 | π / 2 rad | 1.5708 rad |
| 120° | 120π / 180 |
Divide by 60 | 2π / 3 rad | 2.0944 rad |
| 135° | 135π / 180 |
Divide by 45 | 3π / 4 rad | 2.3562 rad |
| 180° (Straight Angle) | 180π / 180 |
Divide by 180 | π rad | 3.1416 rad |
| 270° | 270π / 180 |
Divide by 90 | 3π / 2 rad | 4.7124 rad |
| 360° (Full Circle) | 360π / 180 |
Divide by 180 | 2π rad | 6.2832 rad |
Unit Circle Trigonometry Conversion Table (0° to 360°)
Below is the complete trigonometry unit circle reference table containing standard angle increments across all four quadrants:
| Degrees (°) | Exact π Radians | Decimal Radians (rad) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|---|
| 0° | 0 rad | 0.0000 rad | 0 | 1 | 0 |
| 30° | π/6 rad | 0.5236 rad | 1/2 | √3/2 | √3/3 |
| 45° | π/4 rad | 0.7854 rad | √2/2 | √2/2 | 1 |
| 60° | π/3 rad | 1.0472 rad | √3/2 | 1/2 | √3 |
| 90° | π/2 rad | 1.5708 rad | 1 | 0 | Undefined |
| 120° | 2π/3 rad | 2.0944 rad | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 rad | 2.3562 rad | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 rad | 2.6180 rad | 1/2 | -√3/2 | -√3/3 |
| 180° | π rad | 3.1416 rad | 0 | -1 | 0 |
| 210° | 7π/6 rad | 3.6652 rad | -1/2 | -√3/2 | √3/3 |
| 225° | 5π/4 rad | 3.9270 rad | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 rad | 4.1888 rad | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 rad | 4.7124 rad | -1 | 0 | Undefined |
| 300° | 5π/3 rad | 5.2360 rad | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 rad | 5.4978 rad | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 rad | 5.7596 rad | -1/2 | √3/2 | -√3/3 |
| 360° | 2π rad | 6.2832 rad | 0 | 1 | 0 |
Related Angle and Geometry Calculators
Need to calculate reverse angles or explore trigonometric geometry? Use our connected tools:
- Radians to Degrees Converter — Convert Radians and exact π values back to Degrees.
- Trigonometry Calculator — Compute sine, cosine, tangent, and inverse trigonometric functions.
- Circle Calculator — Calculate arc lengths, sector angles, and circumference.
- Master Conversion Calculator — Convert units across all physical measurement categories.
Frequently Asked Questions (FAQ)
How do you convert 45 degrees to radians in terms of π?
Multiply 45 by π/180: 45π / 180. Divide both by 45: π / 4 radians (approximately 0.7854 rad).
How many radians are in 1 degree?
1 degree is equal to π / 180 ≈ 0.01745329 radians.
Why is a full circle 360 degrees but 2π radians?
360 degrees was chosen by ancient Mesopotamians due to their base-60 counting system and the ~360 days in a solar year. In contrast, 2π radians is mathematically derived from the circle’s circumference formula ($C = 2\pi r$), representing the exact angle needed to wrap the radius around the entire perimeter.