A circular segment — the sliver of a circle cut off by a chord — is fully defined by its radius and central angle, or equivalently by its chord length and height (sagitta). From either pair, this calculator derives the segment's area, chord, height, and arc length instantly, and can solve one representation from the other.
How the Circular Segment Calculator works
A segment's area is the sector's area (the pie-slice bounded by two radii and the arc) minus the triangle formed by those same two radii and the chord: A = ½r²θ − ½r²sinθ = ½r²(θ − sinθ), with θ in radians. The chord follows from the isosceles triangle formed by the two radii and the chord itself, c = 2r·sin(θ/2), and the sagitta — the height of the bulge — is h = r(1 − cos(θ/2)). Arc length is simply s = rθ. All four formulas are exact for any angle between 0° and 360°.
Inputs and what they mean
In "From radius & angle" mode, enter the circle's radius and the central angle in degrees — the angle at the center between the two radii bounding the segment. In "From chord & height" mode, enter the chord length and the sagitta directly; the calculator first back-solves the radius using r = h/2 + c²/(8h), then the central angle using θ = 2·cos⁻¹(1 − h/r), before computing area and arc length the same way. Both modes produce the full set of results — pick whichever pair of measurements you actually have on hand.
Segment vs. sector, and limits
A segment is easy to confuse with a sector: a sector is the pie-slice bounded by two straight radii and the arc, while a segment is bounded by the arc and the straight chord connecting its endpoints — a sector minus its triangle equals a segment. This calculator requires a central angle strictly between 0° and 360°; at 0° there is no segment, and at 360° the "segment" is the entire circle, which the chord/height formulas can't represent (an infinite chord-to-height ratio). It models a flat 2D segment only — for a 3D segment of a sphere or cylinder, additional dimensions are needed.