Finds a circle’s central angle from a known arc length and radius — the angle at the circle’s center that sweeps out that arc.
How it works
The arc length is divided by the radius to give the angle in radians, then converted to degrees.
What this does not include
This does not include finding the arc length from an angle — for that direction, use this site’s arc length calculator instead.
How to use this calculator
- Enter the arc length and the circle’s radius.
A worked example
An arc length of 15.708 on a circle with radius 10: central angle = (arcLength/radius) × (180/π) = 90.0002°.
Arc length 5 on radius 5: central angle = 57.2958° — exactly 1 radian converted to degrees, since arc length equals radius at that point.
What the variables mean
| Variable | Meaning |
|---|---|
| Arc length | Length of the curved segment |
| Radius | Circle’s radius |
Edge cases worth knowing
When arc length equals the radius, the central angle is exactly 1 radian (≈57.2958°) — the defining relationship between radians and arc length, shown directly in the second example.
A radius of zero makes the angle undefined — there’s no meaningful circle to measure an angle within.
Frequently asked questions
Why does dividing arc length by radius give an angle in radians?
Because a radian is defined as the angle where the arc length equals the radius — dividing one by the other directly gives that ratio.
What’s the largest possible central angle?
360°, which corresponds to the full circumference as the “arc length.”
Does the formula change for different circle sizes?
No — the ratio of arc length to radius gives the same angle in radians regardless of how large or small the circle is, since both scale together.
