Arc Length Calculator
Calculate the arc length, radius, or central angle of a circular arc
Distance from center to edge
Angle at the center of the circle
Length along the curved arc
Arc Calculation Results
Arc Length
Radius
Angle
Sector Area
Area of the pie slice
Chord Length
Straight-line distance
Full Circumference
Complete circle length
Angle (Radians)
Angle in radians
Formulas Used
Arc Length
L = rθ
Sector Area
A = ½r²θ
Chord Length
c = 2r·sin(θ/2)
Note: θ must be in radians for these formulas
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About Arc Length Calculator
What is Arc Length?
Arc length is the distance along the curved line that forms part of a circle's circumference. When you trace a portion of a circle's edge between two points, that curved distance is the arc length. It's a fundamental concept in geometry with applications ranging from engineering to astronomy.
How to Use This Calculator
- Select Mode: Choose between finding Arc Length, Radius, or Angle.
- Enter Values: Input the known measurements for your circular arc.
- Select Units: For angles, toggle between Degrees and Radians.
- Review Results: See the calculated values along with related sector and chord data.
Understanding Your Results
Primary Metric
Arc Length: The calculated distance along the curve. This is the main result in finding arc length mode.
Secondary Metrics
Radius: The distance from the center to the circle's edge. Angle: The central angle in degrees or radians. Sector Area: The area of the pie slice enclosed by the arc. Chord Length: The straight-line distance between the arc endpoints.
Arc Length Formulas
Primary Formula
Arc Length = Radius × Angle (in radians)
L = rθ
Where:
- L = Arc length
- r = Radius of the circle
- θ = Central angle in radians
Converting Degrees to Radians
If your angle is in degrees, convert it first:
θ (radians) = θ (degrees) × π/180
Derived Formulas
| Find | Formula |
|---|---|
| Arc Length | L = rθ |
| Radius | r = L/θ |
| Angle | θ = L/r |
Frequently Asked Questions (FAQ)
What is the difference between arc length and chord length?
Arc length is the distance along the curve, while chord length is the straight-line distance between the two endpoints. Arc length is always greater than or equal to chord length.
Why must the angle be in radians for the formula?
The formula L = rθ only works when θ is in radians because radians are defined as the ratio of arc length to radius. This creates the elegant relationship where multiplying radius by angle (in radians) directly gives arc length.
How do I convert between degrees and radians?
- Degrees to radians: Multiply by π/180
- Radians to degrees: Multiply by 180/π
What's the arc length of a full circle?
For a full circle (360° or 2π radians), the arc length equals the circumference: L = 2πr
Can arc length be negative?
No, arc length is always a positive distance. However, angles can be negative (indicating direction), but we typically use the absolute value for arc length calculations.
Important Notes
- Always ensure your angle unit (degrees or radians) is correctly selected
- The radius and arc length must use the same unit of measurement
- For very small angles, arc length closely approximates chord length
Disclaimer: This calculator is for informational purposes only and should not replace professional geometric or engineering advice.