Arc Length Calculator Degrees
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Reviewed by Calculator Editorial Team
Calculating arc length in degrees is essential for geometry, engineering, and physics applications. This calculator provides a simple way to determine the length of an arc when you know the radius and the central angle in degrees.
What is Arc Length?
Arc length is the distance along a curve of a circle. It's calculated based on the central angle (the angle subtended at the center of the circle) and the radius of the circle. The arc length is a straight-line distance measured along the circumference.
In practical terms, arc length helps in determining the length of cables wrapped around a pulley, the distance traveled by a wheel, or the length of a curved road segment.
Arc Length Formula
The formula to calculate arc length (L) when the angle is in degrees is:
L = (θ / 360) × 2πr
Where:
- L = Arc length
- θ = Central angle in degrees
- r = Radius of the circle
- π (pi) ≈ 3.141592653589793
This formula works by first converting the angle from degrees to a fraction of a full circle (360°), then multiplying by the circumference of the circle (2πr).
Note: For angles in radians, the formula is simply L = rθ, where θ is in radians.
How to Use the Calculator
- Enter the radius of the circle in the first input field.
- Enter the central angle in degrees in the second input field.
- Click the "Calculate" button to compute the arc length.
- The result will appear below the calculator with a visual representation.
- Use the "Reset" button to clear all inputs and results.
The calculator will automatically validate your inputs and display an error message if any values are invalid.
Worked Examples
Let's look at a couple of examples to understand how the arc length calculation works.
Example 1: Small Angle
Suppose you have a circle with radius 5 units and a central angle of 30 degrees. What is the arc length?
L = (30 / 360) × 2π × 5
L = (1/12) × 10π
L ≈ 2.61799
The arc length is approximately 2.62 units.
Example 2: Large Angle
For a circle with radius 10 units and a central angle of 180 degrees, what is the arc length?
L = (180 / 360) × 2π × 10
L = 0.5 × 20π
L ≈ 31.4159
The arc length is approximately 31.42 units, which is half the circumference of the circle.
Comparison Table
| Radius (r) | Angle (θ) in degrees | Arc Length (L) |
|---|---|---|
| 5 | 30 | 2.62 |
| 10 | 90 | 5.24 |
| 15 | 180 | 23.56 |
| 20 | 360 | 62.83 |
This table shows how the arc length changes with different radius and angle combinations.
Frequently Asked Questions
What is the difference between arc length and chord length?
Arc length is the distance along the curve of the circle, while chord length is the straight-line distance between two points on the circumference. For small angles, these values are nearly equal, but they diverge as the angle increases.
Can I use this calculator for angles greater than 360 degrees?
Yes, the calculator will work for any angle value. For angles greater than 360 degrees, the calculation will give you the arc length for the equivalent angle within one full rotation (angle modulo 360).
What units should I use for the radius?
The calculator accepts any unit of length for the radius (inches, centimeters, meters, etc.), but the result will be in the same units. Make sure to use consistent units throughout your calculations.
Is there a way to calculate arc length without using degrees?
Yes, if you have the angle in radians, you can use the simpler formula L = rθ. Our calculator can also handle radians if you convert your angle first.