Circle Segment Area Calculator
Find the area of a circular segment, plus its chord and arc length, from the radius and central angle.
How it works
Segment area = (r²/2)(θ − sin θ) · Chord = 2r·sin(θ/2) · Arc = r·θ (θ in radians)
A circular segment is the region between a chord and the arc it cuts off — the part of a circle sliced by a straight line. Its area is the area of the pie-slice sector (r²θ/2) minus the triangle formed by the two radii and the chord (r²·sin θ / 2), which combine into (r²/2)(θ − sin θ). The chord is the straight cut and the arc is the curved boundary of the same segment.
Worked example
For a radius of 5 and a central angle of 90° (θ = π/2 ≈ 1.5708): area = (25/2)(1.5708 − sin 90°) = 12.5 · (1.5708 − 1) ≈ 7.13 units²; chord = 2·5·sin45° ≈ 7.07 units; arc = 5 · 1.5708 ≈ 7.85 units.
Sources: Circular segment — Wolfram MathWorld
Frequently asked questions
What is the difference between a segment and a sector?
A sector is a pie slice bounded by two radii and an arc. A segment is bounded by a chord and an arc — it is the sector with the central triangle removed, so its area is always smaller for the same angle.
Why must the angle be entered in degrees but converted to radians?
The formulas use radians, where the sine term and the arc length behave correctly. The calculator accepts degrees for convenience and converts internally by multiplying by π/180.
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This calculator is for educational purposes. Double-check important results before acting on them.