How this instrument works
A circular segment is the region trapped between a chord and the arc it cuts off — not the whole pie-slice wedge a sector describes, only the crescent left over once the straight-sided triangular part of that wedge is removed. Draw the two radii to the chord's endpoints and a sector appears, area ½r²θ. Inside that sector sits a triangle formed by the same two radii and the chord itself, with area ½r²sinθ by the ordinary two-sides-and-included-angle rule. The segment is what remains once the triangle is subtracted from the sector, which is exactly what A = ½r²(θ − sinθ) computes: sector, minus the wrong-side wedge that the chord fences off.
The subtraction behaves in a way worth sitting with. As θ shrinks toward 0, sinθ tracks θ closely — the two terms nearly cancel — and the gap θ − sinθ collapses roughly as θ³⁄6, a cubic vanishing rather than a straight-line one. On a radius-5 circle, moving the central angle from 0.1 radians to 0.2 radians — doubling it — takes the segment area from about 0.00208 to about 0.01663, close to an eightfold jump rather than a twofold one: thin slivers thin out far faster than their angle suggests, because almost the entire sector at a small angle is already triangle. At the other extreme, θ = π puts the chord straight through the center as a diameter, sinπ = 0, the triangle flattens to a line with zero area, and the segment becomes exactly half the circle.
Outside a classroom this shape shows up whenever something curved is filled partway. The wetted cross-section of a horizontal cylindrical tank or pipe, part-full of liquid, is a circular segment — its area, times the pipe's length, gives the volume sitting inside at that fill depth, which is how tank gauges and partially-full pipe-flow calculations get built. The same shape appears in a stadium arch cut by a horizontal sightline, or a lens ground from a single circular blank.
- Enter the circle's size into the Radius field, using any length unit you like.
- Enter the wedge's sweep into Central angle, choosing degrees, radians, or turns from the unit toggle beside it.
- Read Area for the segment's area, in that same length unit squared.
- Set Central angle to 180° to check the extreme case: Area should land on exactly half the full circle, πr² ⁄ 2.
- Try a small angle, like 5°, to watch Area shrink far faster than the angle itself does.
Worked example — radius 5, angle 1.2 radians
Set Radius to 5 and Central angle to 1.2 radians, close to 68.75°. The sector that angle sweeps out has area ½ × 5² × 1.2 = 12.5 × 1.2 = 15, using the plain sector formula before any correction for the chord straightening one edge of the wedge.
The triangle riding inside that sector — the two radii and the chord, the piece on the wrong side of the cut — has area ½ × 5² × sin(1.2) ≈ 12.5 × 0.9320390859672263 ≈ 11.6504886. Subtracting that from the sector's 15 leaves a segment of about 3.3495114, and Area reports the full-precision figure the sheet actually carries: 3.3495114254096707 — a segment holding barely a fifth of the sector that contains it, because 1.2 radians still cuts off a fairly slender crescent rather than a wide bite of the circle.
Questions
What is the formula for the area of a circular segment?
A = ½r²(θ − sinθ), with θ the central angle in radians. It comes from taking the sector's area, ½r²θ, and subtracting the triangle formed by the two radii and the chord, ½r²sinθ — the segment is whatever the sector leaves behind once that triangular wedge is removed.
How is a segment different from a sector?
A sector is the full pie-slice: two straight radii plus the arc between them, area ½r²θ. A segment is only the curved sliver between the chord and the arc, leaving the triangle out entirely. Every segment sits inside its matching sector; the sector is always the larger of the two, except when θ = 0 and both vanish together.
Why does the triangle's area use sinθ rather than θ itself?
The triangle has two sides of length r meeting at the central angle θ, and the standard area rule for that shape — half the product of two sides times the sine of the included angle — gives ½r²sinθ. That rule, not the sector formula, is where the sine enters; the sector term ½r²θ never had a sine in it to begin with.
Why does segment area shrink so fast as the angle gets small?
Because sinθ closely tracks θ near zero — their difference behaves like θ³⁄6, a cubic term, not a linear one. Halving the central angle roughly divides the segment area by eight rather than two, so a sliver cut at a small angle is thinner than its angle alone would suggest; nearly the whole sector at that angle is already the triangle being subtracted away.
What is the segment area at a 180° central angle?
Exactly half the circle. At θ = π, sinπ = 0, so the triangle term disappears and A = ½r²π = πr² ⁄ 2 — the chord has become a diameter, splitting the disk evenly with no triangular piece left to subtract, since the 'triangle' has flattened onto that same straight diameter line.
Where does circular segment area show up outside geometry class?
In gauging partly filled horizontal cylindrical tanks and pipes: the wetted cross-section at any fill depth is a circular segment, so this formula, multiplied by length, gives the liquid volume at that depth. The same shape describes a lens ground from a circular blank or a stadium roof cut by a horizontal sightline.