Arc Length Angles and Radius W/ Degrees Calculator
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Reviewed by Calculator Editorial Team
Calculating arc length is essential in geometry, engineering, and design. This calculator helps you determine the length of an arc when you know the central angle in degrees and the radius of the circle.
What is Arc Length?
Arc length is the distance along a section of a curve. In the context of a circle, the arc length is the length of the curve that connects two points on the circumference. It's different from the straight-line distance between those two points, which would be the chord length.
The arc length depends on two main factors: the radius of the circle and the central angle that subtends the arc. The central angle is the angle formed by two radii that connect the center of the circle to the two points on the circumference.
Formula
The formula to calculate arc length when the angle is in degrees is:
Where:
- θ is the central angle in degrees
- r is the radius of the circle
- π (pi) is approximately 3.14159
This formula works because a full circle (360 degrees) has a circumference of 2πr. Therefore, the arc length is a fraction of the full circumference corresponding to the fraction of the circle's angle that the arc represents.
How to Calculate Arc Length
To calculate the arc length using this calculator:
- Enter the central angle in degrees in the first input field
- Enter the radius of the circle in the second input field
- Click the "Calculate" button
- The calculator will display the arc length in the same units as the radius
The calculator will also show a visual representation of the circle with the arc highlighted, and provide an explanation of the result.
Example Calculation
Let's say you have a circle with a radius of 10 units and you want to find the arc length for a central angle of 90 degrees.
Using the formula:
Arc Length = 0.25 × 20π
Arc Length ≈ 0.25 × 62.8319
Arc Length ≈ 15.7079 units
So the arc length for a 90-degree angle in a circle with radius 10 units is approximately 15.71 units.
Applications
Calculating arc length has practical applications in various fields:
- Engineering: Designing circular structures, calculating cable lengths, and determining gear tooth profiles
- Architecture: Creating curved designs, calculating window frames, and determining roof shapes
- Manufacturing: Calculating material lengths for circular components and determining cutting paths
- Navigation: Determining distances along curved routes on maps
- Art and Design: Creating precise curved patterns and designs
FAQ
What's 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 the same angle, the arc length will always be longer than the chord length.
Can I use this calculator for angles greater than 360 degrees?
No, this calculator is designed for angles between 0 and 360 degrees. For angles greater than 360 degrees, you would need to calculate the full circumference and multiply by the number of complete rotations.
What if I have the angle in radians instead of degrees?
The formula for arc length when the angle is in radians is simply θ × r. You can convert radians to degrees by multiplying by 180/π, or use our radians to degrees converter.