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° |
Radians and degrees are the two primary units used globally to express angles, rotational motion, arc lengths, and trigonometric functions. While the degree (°) is the intuitive unit used in elementary geometry, navigation, cartography, and everyday engineering (dividing a full circle into 360 equal parts), the radian (rad) is the fundamental SI-derived unit used across calculus, physics, computer graphics, and advanced mathematics.
Our free online Radians to Degrees Converter provides instant, high-precision angle conversions in both directions:
- Radians to Degrees: Multiply radians by
180 / π(approximately57.29577951). - Degrees to Radians: Multiply degrees by
π / 180(approximately0.01745329).
Exact Angle Conversion Formulas
Because a full 360-degree circle forms a circumference of 2πr on a unit circle, 2π radians equals 360 degrees (or π radians = 180 degrees). This establishes exact mathematical relationships:
| Conversion Direction | Exact Algebraic Formula | Decimal Multiplier (Approximate) |
|---|---|---|
| Radians to Degrees | Degrees = Radians × (180 / π) |
Degrees = Radians × 57.29577951 |
| Degrees to Radians | Radians = Degrees × (π / 180) |
Radians = Degrees × 0.01745329 |
Step-by-Step Calculation Example
To explicitly convert 15 radians to degrees, you must multiply the radian value by the standard conversion factor.
| Math Step | Execution Logic | Calculated Value |
|---|---|---|
| Step 1: Input Radians | Identify the starting radian value. | 15 rad |
| Step 2: Conversion Factor | Select the radians-to-degrees multiplier. | 57.29577951 |
| Step 3: Final Multiplication | 15 × 57.29577951 |
859.44° |
Essential Unit Circle Trigonometry Benchmarks
In calculus and trigonometry, standard angles are expressed as exact fractions of π radians:
| Exact Pi Radians | Decimal Radians (rad) | Degrees (°) | Geometric Angle Description |
|---|---|---|---|
| 0 rad | 0.0000 rad | 0 ° | Zero angle / positive x-axis origin |
| π / 6 rad | 0.5236 rad | 30 ° | Acute 30-degree angle |
| π / 4 rad | 0.7854 rad | 45 ° | Isosceles right triangle angle |
| π / 3 rad | 1.0472 rad | 60 ° | Equilateral triangle interior angle |
| 1.0 rad | 1.0000 rad | 57.2958 ° | 1 Radian exact arc-length benchmark |
| π / 2 rad | 1.5708 rad | 90 ° | Right angle / perpendicular lines |
| π rad | 3.1416 rad | 180 ° | Straight angle / half revolution |
| 3π / 2 rad | 4.7124 rad | 270 ° | Three-quarter revolution |
| 2π rad | 6.2832 rad | 360 ° | Full circle rotation |
Radian to Degree Conversion Table
Below is a quick reference lookup table converting decimal radians to degrees from 0.01 rad up to 1,000 rad:
| Radians (rad) | Exact Equivalent in Degrees (°) | Rounded Degrees (2 Decimals) |
|---|---|---|
| 0.01 rad | 0.57295780 ° | 0.57 ° |
| 0.10 rad | 5.72957795 ° | 5.73 ° |
| 1.00 rad | 57.29577951 ° | 57.30 ° |
| 2.00 rad | 114.59155903 ° | 114.59 ° |
| 3.00 rad | 171.88733854 ° | 171.89 ° |
| 5.00 rad | 286.47889757 ° | 286.48 ° |
| 10.00 rad | 572.95779513 ° | 572.96 ° |
| 20.00 rad | 1,145.91559026 ° | 1,145.92 ° |
| 50.00 rad | 2,864.78897565 ° | 2,864.79 ° |
| 100.00 rad | 5,729.57795131 ° | 5,729.58 ° |
| 1,000.00 rad | 57,295.77951308 ° | 57,295.78 ° |
History & Mathematics: Radian vs. Degree
The Origin of the Radian (rad)
Measuring angles via arc length on a unit circle was introduced by mathematicians around 1400. English mathematician Roger Cotes described the concept of the radian in 1714. However, the specific term “radian” was first coined in print in 1873 by James Thomson (brother of Lord Kelvin). Because a radian is defined as the ratio of arc length to radius (s / r), the units cancel out, making the radian a dimensionless quantity in physics and calculus.
The Origin of the Degree (°)
The origin of dividing a circle into 360 degrees traces back to ancient Babylonian astronomy. The number 360 was chosen because it closely matched the 365-day solar year calendar and is a highly composite number with 24 divisors (divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360).
To convert angles to radians, check our Degrees to Radians Converter. For right-triangle trigonometry and circular arc calculations, visit our Right Triangle Calculator or Circle Calculator. For general unit transformations, explore our Conversion Calculator.
Frequently Asked Questions (FAQ)
How many degrees are in 1 radian?
There are approximately 57.29578 degrees in 1 radian (180 / π).
How many radians are in 180 degrees?
There are exactly π radians (approximately 3.14159265 rad) in 180 degrees.
Why do calculus and physics prefer radians over degrees?
Radians make calculus derivatives mathematically elegant. For example, d/dx(sin x) = cos x is only true when x is measured in radians. If measured in degrees, an awkward scale factor of π/180 must be multiplied onto every derivative.
How do I convert π/4 radians to degrees?
Simply replace π with 180 degrees: 180 / 4 = 45 degrees.