Convert Arc Length to Degrees Calculator
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Reviewed by Calculator Editorial Team
Convert arc length measurements to degrees with this precise calculator. Whether you're working with circles, sectors, or circular paths, this tool provides accurate degree conversions based on the arc length and radius of the circle.
How to Use This Calculator
Using our arc length to degrees converter is straightforward:
- Enter the arc length in your chosen units (meters, centimeters, inches, etc.)
- Enter the radius of the circle in the same units
- Click "Calculate" to get the equivalent angle in degrees
- Review the result and chart visualization
- Use the "Reset" button to start a new calculation
The calculator handles all unit conversions internally, so you can work with any consistent measurement system.
The Formula Explained
The relationship between arc length (L), radius (r), and the central angle (θ) in degrees is given by:
Formula
θ (degrees) = (L / r) × (180 / π)
Where:
- θ is the central angle in degrees
- L is the arc length
- r is the radius of the circle
- π (pi) is approximately 3.14159265359
This formula converts the ratio of arc length to radius (which is in radians) to degrees by multiplying by 180/π.
Worked Examples
Example 1: Simple Circle
Given a circle with radius 5 meters and an arc length of 3 meters:
Calculation
θ = (3 / 5) × (180 / π) ≈ 103.6396°
This means the arc spans approximately 103.64 degrees of the circle.
Example 2: Engineering Application
An engineer measures a 12-inch arc on a 6-inch radius wheel:
Calculation
θ = (12 / 6) × (180 / π) ≈ 180°
The result shows this arc represents exactly half of the wheel's circumference.
| Arc Length (m) | Radius (m) | Angle (degrees) |
|---|---|---|
| 1.5 | 2.0 | 135.00 |
| 3.0 | 4.0 | 135.00 |
| 2.0 | 1.0 | 114.59 |
This table shows how different arc lengths with the same angle ratio (L/r) result in the same angle.
Practical Applications
Converting arc length to degrees has numerous practical applications:
- Engineering: Calculating gear tooth angles and wheel rotations
- Architecture: Designing circular structures and domes
- Navigation: Determining compass bearings from arc measurements
- Robotics: Planning circular paths for robotic arms
- Art and Design: Creating precise circular patterns
Important Note
For very small angles, you may need to use radians instead of degrees. Our calculator automatically provides both values when appropriate.