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° |
Degrees and radians are the two fundamental angular measurement units used across geometry, calculus, physics, robotics, computer graphics programming, navigation, and civil engineering. While the degree (°) is the common everyday unit based on dividing a full circle turn into 360 equal parts, the radian (rad) is the SI-derived natural unit of angle measurement based on the mathematical constant π (pi) and circle arc length.
Our free online Degrees to Radians Converter provides instant, high-precision angular conversions in both directions:
- Degrees to Radians: Multiply degrees by
π / 180(or multiply by0.0174532925). - Radians to Degrees: Multiply radians by
180 / π(or multiply by57.2957795).
Exact Conversion Formulas & Pi Relationships
A complete 360-degree rotation of a circle is mathematically equivalent to an arc length of 2π radians around a unit circle. This establishes exact mathematical relationships:
| Conversion Direction | Exact Algebraic Formula | Decimal Equivalent (5 Decimal Places) |
|---|---|---|
| Degrees to Radians | Radians = Degrees × (π / 180) |
Radians = Degrees × 0.01745 |
| Radians to Degrees | Degrees = Radians × (180 / π) |
Degrees = Radians × 57.29578 |
Step-by-Step Calculation Example
To convert 15 degrees to radians:
15° × (π / 180) = π / 12 = 0.26179939 radians
Thus, 15° is equal to exactly π/12 radians (approximately 0.2618 rad).
Trigonometric Unit Circle Conversion Chart (Exact Pi Fractions)
In trigonometry, calculus, and physics, standard angles around the unit circle are expressed as exact fractions of π:
| Degree Measure (°) | Exact Pi Fraction (Radians) | Decimal Radians (rad) | Trigonometric Angle Name / Position |
|---|---|---|---|
| 0° | 0 rad | 0.000000 rad | Zero angle / Positive X-axis |
| 30° | π / 6 rad | 0.523599 rad | 1/12th rotation (30-60-90 triangle angle) |
| 45° | π / 4 rad | 0.785398 rad | 1/8th rotation (Isosceles right triangle angle) |
| 60° | π / 3 rad | 1.047198 rad | 1/6th rotation (Equilateral triangle angle) |
| 90° | π / 2 rad | 1.570796 rad | Right angle / Positive Y-axis (Quarter turn) |
| 180° | π rad | 3.141593 rad | Straight angle / Negative X-axis (Half turn) |
| 270° | 3π / 2 rad | 4.712389 rad | Negative Y-axis (Three-quarter turn) |
| 360° | 2π rad | 6.283185 rad | Full turn rotation (1 complete circle) |
Degree to Radian Conversion Table
Below is a handy reference lookup table converting decimal degree measurements from 0.01° up to 1,000°:
| Degrees (°) | Radians (rad) | Mathematical & Arc Length Benchmark |
|---|---|---|
| 0.01 ° | 0.000175 rad | 174.5 micro-radians |
| 0.10 ° | 0.001745 rad | 1.745 milli-radians |
| 1.00 ° | 0.017453 rad | 1.0 Degree Exact Benchmark (π/180 = 0.017453 rad) |
| 2.00 ° | 0.034907 rad | π/90 radians |
| 5.00 ° | 0.087266 rad | π/36 radians |
| 10.00 ° | 0.174533 rad | π/18 radians |
| 57.30 ° (57.2958°) | 1.000000 rad | 1.0 Radian Exact Arc Length Benchmark (180/π ≈ 57.30°) |
| 100.00 ° | 1.745329 rad | 5π/9 radians |
| 180.00 ° | 3.141593 rad | π radians exact (Half circle turn) |
| 360.00 ° | 6.283185 rad | 2π radians exact (Full circle turn) |
| 720.00 ° | 12.566371 rad | 4π radians (2 complete rotations) |
| 1,000.00 ° | 17.453293 rad | 50π/9 radians |
History & Mathematics: Babylonian 360° vs. Roger Cotes’ Radian
The Origin of the 360-Degree Circle
The division of a circle into 360 degrees dates back over 4,000 years to ancient Mesopotamia and Babylon. Historians believe Babylonian astronomers chose 360 because their calendar year contained roughly 360 days and their base-60 (sexagesimal) numbering system favoured 360, which has 24 clean integer divisors (divisible by 1 through 10, except 7).
Roger Cotes and the Discovery of the Radian (1714)
Measuring angles by arc length was used by mathematicians as early as 1400. However, the concept of the radian is credited to English mathematician Roger Cotes in 1714, who defined 1 radian as the angle subtended at the center of a circle by an arc equal in length to the circle’s radius (arc length s = radius r). In 1873, James Thomson coined the term “radian” in print. Radians are dimensionless ratios, making them essential for calculus derivatives (e.g., d/dx(sin x) = cos x only holds true when x is in radians!).
To convert radians back to degrees, visit our Radians to Degrees Converter. For geometric circle properties, check our Circle Calculator or explore our Conversion Calculator.
Frequently Asked Questions (FAQ)
How many radians are in 1 degree?
There are exactly π / 180 radians in 1 degree (approximately 0.01745329 rad).
How many degrees are in 1 radian?
There are approximately 57.29578 degrees in 1 radian (180 / π).
What is 90 degrees in radians?
90 degrees equals 90 × π/180 = π/2 radians (approximately 1.5708 rad).
What is 180 degrees in radians?
180 degrees equals exactly π radians (approximately 3.14159 rad).
Why do calculus and physics prefer radians over degrees?
Because radians are derived naturally from the circle arc ratio (s = r · θ), eliminating ugly conversion constants from calculus derivatives like d/dx(sin x) = cos x and physics equations for angular velocity (v = ωr).