Calculate a circle's central angle from arc length, chord length, or sector area with live validation.
Last updated: June 2026 | By Patchworkr Team
Live results update as you type
Radius needs a real number.
A central angle has its vertex at the center of a circle. It determines the arc, the sector, and the chord it intercepts.
The inverse formulas are exact: arc length gives theta by s / r, sector area gives theta by 2A / r^2, and chord length gives theta by 2 asin(c / 2r).
The calculator keeps the domain checks explicit so invalid circle states are rejected before a result is shown.
For a circle of radius 10 and arc length 10.4719755, theta = 10.4719755 / 10 = 1.04719755 radians.
The slice is exactly one sixth of the full circle, so the outputs line up with the arc-length relationship.
Can the chord be longer than the diameter?
No. The calculator rejects that case because the chord would not fit inside the circle.
Why convert degrees to radians?
The inverse formulas for arc and sector use radians directly, so the tool converts internally.
What if the sector area is zero?
That is a degenerate boundary case and gives an angle of zero.
Does the radius need to be positive?
Yes. A non-positive radius is not a valid circle for this calculator.
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