Radians to Degrees Converter
Convert angle measurements from radians to degrees quickly and accurately. Supports decimal values, fractions, and expressions with pi (π). Get results in both decimal degrees and degrees-minutes-seconds format.
Enter a decimal number (e.g., 1.5708 for π/2)
Common Radians to Degrees Conversions
| Radians (Expression) | Radians (Decimal) | Degrees (°) |
|---|---|---|
| 0 | 0 | 0° |
| π/12 | 0.261799 | 15° |
| π/6 | 0.523599 | 30° |
| π/4 | 0.785398 | 45° |
| π/3 | 1.047198 | 60° |
| π/2 | 1.570796 | 90° |
| 2π/3 | 2.094395 | 120° |
| 3π/4 | 2.356194 | 135° |
| 5π/6 | 2.617994 | 150° |
| π | 3.141593 | 180° |
| 3π/2 | 4.712389 | 270° |
| 2π | 6.283185 | 360° |
How to Convert Radians to Degrees
Converting radians to degrees is essential for working with angles in various applications. While radians are the standard unit in mathematics and calculus, degrees are more intuitive for everyday use and practical applications like navigation, construction, and engineering.
The Conversion Formula
The formula to convert radians to degrees is based on the relationship that π radians equals 180 degrees:
Since 180/π ≈ 57.29577951, you can also use this simplified formula:
This means that 1 radian is approximately equal to 57.2958 degrees.
Example 1: Converting π/2 Radians to Degrees
Using the formula:
degrees = (π/2) × (180 / π) = 180/2 = 90°
Using decimal:
degrees = 1.5708 × 57.29578 = 90°
Example 2: Converting 2 Radians to Degrees
Using the formula:
degrees = 2 × (180 / π) = 360/π ≈ 114.5916°
Using decimal multiplier:
degrees = 2 × 57.29578 ≈ 114.5916°
Understanding Radians
A radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. This definition makes radians a "natural" unit for measuring angles because it directly relates the angle to the geometry of the circle.
Key radian values to understand:
- 0 radians = 0° (no rotation)
- π/2 radians ≈ 1.5708 rad = 90° (quarter turn)
- π radians ≈ 3.1416 rad = 180° (half turn)
- 3π/2 radians ≈ 4.7124 rad = 270° (three-quarter turn)
- 2π radians ≈ 6.2832 rad = 360° (full rotation)
Visual representation of common angles in both radians and degrees
Degrees-Minutes-Seconds (DMS) Format
Angles in degrees can also be expressed in degrees-minutes-seconds (DMS) format, which is commonly used in navigation, surveying, and cartography.
The conversion works as follows:
- 1 degree = 60 minutes of arc (')
- 1 minute = 60 seconds of arc (")
- Therefore, 1 degree = 3,600 seconds
Example: Converting 45.5° to DMS Format
1. The whole number part is the degrees: 45°
2. Multiply the decimal by 60: 0.5 × 60 = 30 minutes
3. If there's a decimal in minutes, multiply by 60 for seconds
4. Final result: 45° 30' 0"
Why Understanding This Conversion Matters
Programming and Software Development: Most programming languages use radians for trigonometric functions. When working with user input in degrees, you must convert to radians before calculations, then convert back to degrees for display.
Physics and Engineering: While physics often uses radians (especially for angular velocity and acceleration), engineering drawings and specifications frequently use degrees. Being able to convert between them is essential.
Navigation and GPS: Geographic coordinates use degrees, but some calculations (like great circle distances) are easier in radians. Understanding the conversion allows for accurate navigation calculations.
Computer Graphics and Animation: 3D graphics often require rotation matrices that work in radians, while artists and designers think in degrees. Game engines typically accept either unit but perform calculations in radians internally.
Common Mistakes to Avoid
- Forgetting to convert: Always check if your calculator or programming language expects radians or degrees for trigonometric functions.
- Rounding errors: π is irrational, so conversions will have small rounding errors. Use sufficient precision for your application.
- Confusing the formulas: Remember: multiply by 180/π to convert TO degrees, multiply by π/180 to convert FROM degrees.
Quick Reference: The Key Relationship
The fundamental relationship to remember is:
From this single fact, you can derive all radian-to-degree and degree-to-radian conversions.