Arc Distance (Arc Length) Calculator
Use this calculator to find the distance along a circle between two radii (the arc length).
If your angle is in degrees, convert first: θrad = θdeg × π / 180
Tip: The calculator treats negative angles as positive distance.
What Is Arc Distance?
Arc distance, also called arc length, is the distance measured along the curved edge of a circle. Unlike a straight-line distance (like a chord), arc distance follows the circle’s circumference.
If you know the radius of the circle and the central angle, you can calculate arc distance quickly. This is useful in geometry, engineering, construction, robotics, wheel rotation problems, and even map projections.
The Core Formula
When the angle is in radians
The formula is:
s = rθ
- s = arc distance (arc length)
- r = radius of the circle
- θ = central angle in radians
When the angle is in degrees
Convert degrees to radians first:
θrad = θdeg × π / 180
Then apply s = rθ.
Quick Example
Suppose a circle has radius 10 and the central angle is 90°.
- Convert to radians: 90 × π / 180 = π/2
- Apply formula: s = 10 × (π/2) = 5π
- Decimal value: s ≈ 15.7079
So the arc distance is approximately 15.71 units.
Why Arc Distance Matters
1) Engineering and manufacturing
When designing curved parts, tracks, or rotating mechanisms, arc measurements help define material length and movement paths.
2) Motion and rotation
Wheel travel distance is based on arc relationships. In rotational systems, angle and radius determine linear displacement.
3) Architecture and construction
Arches, circular walkways, and curved railings often require arc length to estimate cuts and material quantities.
4) Math and physics education
Arc distance appears in trigonometry, calculus, angular velocity problems, and unit-circle analysis.
Common Mistakes to Avoid
- Forgetting unit conversion: using degrees directly in s = rθ without converting to radians.
- Using diameter instead of radius: radius is half the diameter.
- Expecting negative length: physical distance is non-negative; angle direction is usually handled separately.
- Unit mismatch: if radius is in centimeters, arc distance will be in centimeters.
Related Circle Quantities (Also Calculated Above)
Besides arc distance, the calculator also returns:
- Chord length — straight-line distance between arc endpoints.
- Sector area — area enclosed by the arc and two radii.
- Fraction of a full circle — what portion of 360° (or 2π radians) your angle represents.
FAQ
Is arc distance the same as circumference?
Not always. Circumference is the full boundary of the circle. Arc distance is only part of that boundary unless the angle is 360° (or 2π radians).
Can I use radians directly?
Yes. In fact, radians are the natural unit for the formula s = rθ.
What if the angle is bigger than 360°?
The calculation still works. It represents one or more full turns plus additional arc.
Final Notes
Arc distance is one of the most practical geometry calculations. With just a radius and an angle, you can model curved motion, estimate materials, and solve many real-world design problems quickly and accurately.