Segment = sector − triangle = ½r²(θ − sin θ). For a unit circle and a 90° angle: the sector is π/4 ≈ 0.785, the triangle inside it is 0.5, so the segment is about 0.285. Quoting the sector would overstate it by nearly two thirds.
Segment or sector
Two different pieces of a circle, and the difference is a triangle.
A sector is the pie slice: bounded by two radii and the arc between them.
A segment is what a straight cut leaves: bounded by a chord and the arc. It is the sector with the triangle between the two radii removed.
The gap is large at small angles. At 30° the segment is only about 2% of the sector — the triangle takes nearly all of it. At 180° they are equal, since the triangle has collapsed to nothing.
So using the sector formula for a segment is not a small error. It is a large one for exactly the shallow cuts people most often want to measure, and this page shows both numbers so the subtraction is visible.
The sagitta
The sagitta is the height of the segment: the distance from the middle of the chord out to the arc. The name is Latin for arrow, and the picture is a bow with its string — the chord is the string, the arc is the bow, and the sagitta is the arrow.
It is the easiest of these quantities to measure in practice. Lay a straight edge across an arc and measure the gap in the middle: that is the sagitta, and it needs no access to the centre.
Which makes it the practical way to find the radius of a large arc. From a chord of length c and a sagitta s, the radius is (c²/4 + s²)/(2s) — and this page inverts that for you if you enter the height.
Railway engineers used exactly this to check track curvature, with a string across the rail and a ruler at the midpoint. Optical shops use it to measure lens curvature, where the instrument is called a spherometer.
Working backwards
The central angle is the natural input for the formulas and the least natural thing to measure. So this page accepts all three of the quantities you might actually have.
From the chord: θ = 2·arcsin(c/2r). A chord longer than the diameter is refused, because no circle of that radius contains it.
From the height: θ = 2·arccos(1 − h/r).
All three routes convert to an angle first and then run the same formulas, so they cannot disagree with each other. Entering a chord and reading back the angle, then entering that angle, gives the same segment.
One asymmetry worth knowing: a given chord fits two segments — the minor one and the major one on the other side of it. The chord alone does not say which, and this page returns the minor. A height above the radius unambiguously means the major one.
Past a half circle
Above 180° the segment is the larger piece, containing the centre, and the formula keeps working without modification.
It works because sin θ goes negative past 180°, so −sin θ becomes positive and the triangle term adds to the sector rather than subtracting from it. That is the correct behaviour: the triangle now lies outside the region being measured.
A useful check, and one this page’s test suite asserts across every angle: a segment and the segment on the other side of the same chord always add up to the whole circle.
At exactly 180° the two coincide, both being semicircles, and the triangle has no area at all.
Where it gets used
Tank contents. A horizontal cylindrical tank holds a segment of liquid, so the volume is the segment area times the length. It is why a dipstick on a round tank is not linear.
Civil engineering. Cuttings, culverts and arch bridges are segments, and the chord-and-sagitta method measures curvature without access to the centre.
Optics. A lens surface is a spherical segment, and a spherometer measures its sagitta to find the radius.
Geometry problems. The area between a chord and an arc is a standard exam question, and the standard wrong answer is the sector.
Sources and methodology
The segment formulas are standard geometry; these are the references.
Method. The three input routes — angle, chord and height — all convert to a central angle first, so they cannot disagree with each other. Both the sector and the triangle are computed and reported alongside the segment rather than only their difference, because the commonest error on this topic is quoting one for the other. Segment areas are checked against numerical integration under the circle, with the major-segment case handled separately since integrating above the chord always yields the minor one. That engine is verified on every change against 114 hand-written assertions, including that a segment and its complement always sum to the whole circle across every angle from 5° to 355°, and that segment areas match direct numerical integration under the circle. The count and the per-case breakdown are published on the formula verification page.
Read the guide
For the whole circle rather than a piece of it, the Circle Calculator works from any single measurement.