Degrees to Radians Calculator
Online calculator and formula for converting degrees to radians
Degrees to Radians Calculator
Instructions
Enter the angle in degrees and click Calculate. The calculator automatically converts the value to radians.
Conversion - Overview
What are Degrees and Radians?
Degrees (°) and Radians (rad) are two different units of measurement for angles.
- Full circle in degrees: 360°
- Full circle in radians: 2π ≈ 6.283 rad
Conversion Formula
The formula to convert degrees to radians:
\(\displaystyle \text{rad} = \frac{\text{deg} \cdot \pi}{180°} \)
where π (Pi) ≈ 3.14159265359
Important Angle Values
| Degrees | Radians |
|---|---|
| 0° | 0 rad |
| 30° | π/6 ≈ 0.524 rad |
| 45° | π/4 ≈ 0.785 rad |
| 60° | π/3 ≈ 1.047 rad |
| 90° | π/2 ≈ 1.571 rad |
| 180° | π ≈ 3.142 rad |
| 360° | 2π ≈ 6.283 rad |
Reverse Conversion
Converting radians to degrees:
\(\displaystyle \text{deg} = \frac{\text{rad} \cdot 180°}{\pi} \)
Description of Converting Degrees to Radians
Fundamentals
Degrees and radians are two different units for measuring angles. While degrees are the most common unit in everyday life, radians are frequently used in mathematics, physics, and engineering sciences.
Definition of Radians:
A radian is the angle at which the arc length equals the radius of the circle.
\(\displaystyle 1 \text{ rad} = \frac{180°}{\pi} \approx 57.2958° \)
Formula Derivation
A full circle has 360° or 2π radians. This gives us the proportional equation:
\(\displaystyle 360° = 2\pi \text{ rad} \)
\(\displaystyle 180° = \pi \text{ rad} \)
\(\displaystyle 1° = \frac{\pi}{180°} \text{ rad} \)
By multiplying by the degree value, we get the conversion formula:
\(\displaystyle \text{rad} = \text{deg} \cdot \frac{\pi}{180°} \)
Detailed Examples
Example 1: 90° to Radians
Given:
Angle = 90°
Calculation:
\(\displaystyle \text{rad} = \frac{90° \cdot \pi}{180°} = \frac{\pi}{2} \approx 1.571 \text{ rad} \)
Result: 90° equals π/2 rad or approximately 1.571 rad
Example 2: 45° to Radians
Given:
Angle = 45°
Calculation:
\(\displaystyle \text{rad} = \frac{45° \cdot \pi}{180°} = \frac{\pi}{4} \approx 0.785 \text{ rad} \)
Result: 45° equals π/4 rad or approximately 0.785 rad
Example 3: 360° to Radians
Given:
Angle = 360° (full circle)
Calculation:
\(\displaystyle \text{rad} = \frac{360° \cdot \pi}{180°} = 2\pi \approx 6.283 \text{ rad} \)
Result: 360° equals 2π rad or approximately 6.283 rad
Example 4: 30° to Radians
Given:
Angle = 30°
Calculation:
\(\displaystyle \text{rad} = \frac{30° \cdot \pi}{180°} = \frac{\pi}{6} \approx 0.524 \text{ rad} \)
Result: 30° equals π/6 rad or approximately 0.524 rad
Advantages of Radians
- Mathematical simplicity: Many formulas become simpler
- Natural unit: Direct relationship to the circumference
- Derivatives: Simpler differential calculus for trigonometric functions
- Dimensionless: Radians are a ratio of two lengths
- Scientific standard: Preferred unit in physics and higher mathematics
Practical Applications
- Mathematics: Calculus, trigonometry, differential equations
- Physics: Circular motion, oscillations, waves
- Engineering: Mechanics, electrical engineering, control systems
- Computer graphics: Rotations, animations, 3D transformations
- Navigation: Course calculations, GPS systems
- Astronomy: Planetary orbits, angle calculations
- Programming: Mathematical libraries use radians
Extended Conversion Table
| Degrees (°) | Radians (exact) | Radians (decimal) | Angle Type |
|---|---|---|---|
| 0° | 0 | 0 | Zero angle |
| 30° | π/6 | ≈ 0.524 | Acute angle |
| 45° | π/4 | ≈ 0.785 | Acute angle |
| 60° | π/3 | ≈ 1.047 | Acute angle |
| 90° | π/2 | ≈ 1.571 | Right angle |
| 120° | 2π/3 | ≈ 2.094 | Obtuse angle |
| 135° | 3π/4 | ≈ 2.356 | Obtuse angle |
| 180° | π | ≈ 3.142 | Straight angle |
| 270° | 3π/2 | ≈ 4.712 | Reflex angle |
| 360° | 2π | ≈ 6.283 | Full angle |
Important Note
Calculator settings: When using calculators or software programs, make sure they are operating in degree mode (DEG) or radian mode (RAD). Wrong settings lead to incorrect results!
Programming: Most programming languages (C, C++, Java, Python, etc.) expect angles in radians, not degrees. A conversion is therefore often necessary.
Relationship with Arc Length
Calculate arc length:
\(\displaystyle s = r \cdot \alpha \)
where s is the arc length, r is the radius, and α is the angle in radians
Circular sector area:
\(\displaystyle A = \frac{1}{2} r^2 \cdot \alpha \)
where A is the area of the circular sector