Radians to Degrees Calculator
Online calculator and formula for converting radians to degrees
Radians to Degrees Calculator
Instructions
Enter the angle in radians and click Calculate. The calculator automatically converts the value to degrees.
Conversion - Overview
What are Radians and Degrees?
Radians (rad) and Degrees (°) are two different units of measurement for angles.
- Full circle in radians: 2π ≈ 6.283 rad
- Full circle in degrees: 360°
Conversion Formula
The formula to convert radians to degrees:
\(\displaystyle \text{deg} = \frac{\text{rad} \cdot 180°}{\pi} \)
where π (Pi) ≈ 3.14159265359
Important Angle Values
| Radians | Degrees |
|---|---|
| 0 rad | 0° |
| π/6 ≈ 0.524 rad | 30° |
| π/4 ≈ 0.785 rad | 45° |
| π/3 ≈ 1.047 rad | 60° |
| π/2 ≈ 1.571 rad | 90° |
| π ≈ 3.142 rad | 180° |
| 2π ≈ 6.283 rad | 360° |
Reverse Conversion
Converting degrees to radians:
\(\displaystyle \text{rad} = \frac{\text{deg} \cdot \pi}{180°} \)
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Description of Converting Radians to Degrees
Fundamentals
Radians and degrees are two different units for measuring angles. Radians are commonly used in mathematics, physics, and engineering, while degrees are the most common unit in everyday life.
Definition of Degrees:
A degree is 1/360th of a full circle. This division comes from Babylonian mathematics and is based on the sexagesimal system.
\(\displaystyle 1° = \frac{\pi}{180°} \text{ rad} \approx 0.01745 \text{ rad} \)
Formula Derivation
A full circle equals 2π radians or 360°. This gives us the proportional equation:
\(\displaystyle 2\pi \text{ rad} = 360° \)
\(\displaystyle \pi \text{ rad} = 180° \)
\(\displaystyle 1 \text{ rad} = \frac{180°}{\pi} \)
By multiplying by the radian value, we get the conversion formula:
\(\displaystyle \text{deg} = \text{rad} \cdot \frac{180°}{\pi} \)
Detailed Examples
Example 1: π/2 rad to degrees
Given:
Angle = π/2 rad ≈ 1.571 rad
Calculation:
\(\displaystyle \text{deg} = \frac{\frac{\pi}{2} \cdot 180°}{\pi} = \frac{180°}{2} = 90° \)
Result: π/2 rad equals 90°
Example 2: 1 radian to degrees
Given:
Angle = 1 rad
Calculation:
\(\displaystyle \text{deg} = \frac{1 \cdot 180°}{\pi} \approx \frac{180°}{3.14159} \approx 57.2958° \)
Result: 1 rad equals approximately 57.3°
Example 3: 2π rad to degrees
Given:
Angle = 2π rad ≈ 6.283 rad (full circle)
Calculation:
\(\displaystyle \text{deg} = \frac{2\pi \cdot 180°}{\pi} = 2 \cdot 180° = 360° \)
Result: 2π rad equals 360° (full circle)
Example 4: π/4 rad to degrees
Given:
Angle = π/4 rad ≈ 0.785 rad
Calculation:
\(\displaystyle \text{deg} = \frac{\frac{\pi}{4} \cdot 180°}{\pi} = \frac{180°}{4} = 45° \)
Result: π/4 rad equals 45°
When to Use Degrees?
- Everyday applications: Navigation, geodesy, cartography
- Civil engineering: Angles of inclination, roof pitches
- Surveying: Land surveying, angle measurements
- Geography: Longitude and latitude
- Astronomy: Celestial coordinates (partially)
- Meteorology: Wind directions
- Daily life: Compass, orientation, angle specifications
When to Use Radians?
- Mathematics: Calculus, trigonometry, differential equations
- Physics: Circular motion, rotational dynamics, waves
- Engineering: Mechanics, electrical engineering
- Computer graphics: 3D transformations, rotations
- Programming: Mathematical functions (sin, cos, tan)
- Robotics: Joint angles, motion control
- Science: Precise calculations, research
Extended Conversion Table
| Radians (exact) | Radians (decimal) | Degrees (°) | Angle Type |
|---|---|---|---|
| 0 | 0 | 0° | Zero angle |
| π/6 | ≈ 0.524 | 30° | Acute angle |
| π/4 | ≈ 0.785 | 45° | Acute angle |
| π/3 | ≈ 1.047 | 60° | Acute angle |
| π/2 | ≈ 1.571 | 90° | Right angle |
| 2π/3 | ≈ 2.094 | 120° | Obtuse angle |
| 3π/4 | ≈ 2.356 | 135° | Obtuse angle |
| π | ≈ 3.142 | 180° | Straight angle |
| 3π/2 | ≈ 4.712 | 270° | Reflex angle |
| 2π | ≈ 6.283 | 360° | Full angle |
Important Note
Programming: Most programming languages (C, C++, Java, Python, JavaScript, etc.) return trigonometric functions in radians, not degrees. If you want to work with degrees, you must convert the result using this formula.
Calculators: Make sure your calculator is in degree mode (DEG) or radian mode (RAD). Wrong settings lead to incorrect results!
Relationship with Arc Length
Arc length from radians:
\(\displaystyle s = r \cdot \alpha \)
where s is the arc length, r is the radius, and α is the angle in radians
Arc length from degrees:
\(\displaystyle s = \frac{r \cdot \alpha \cdot \pi}{180°} \)
where α is the angle in degrees
Practical Tips
- Memory aid: 1 radian ≈ 57.3° (exactly: 180°/π)
- Quick calculation: For small angles, approximately: 1 rad ≈ 60°
- Excel/Spreadsheets: Use the function =DEGREES(Radians)
- Python:
import math; degrees = math.degrees(radians) - JavaScript:
degrees = radians * (180 / Math.PI)