Circular Segment Area Calculator
Find the area of a circular segment — the region between a chord and its arc — from the radius and central angle in degrees.
How it works
Area = ½ · r² · (θ − sin θ) · θ in radians = angle · π/180
A circular segment is what remains when you slice a circle with a straight chord — the region between that chord and the arc above it. Its area equals the sector’s area minus the triangle formed by the two radii and the chord, which simplifies to ½·r²·(θ − sin θ) with θ in radians. Because sin θ is subtracted, the segment is always smaller than the corresponding sector.
Worked example
For a radius of 5 and a 90° angle, θ = π/2 ≈ 1.5708, so area = ½·25·(1.5708 − sin 1.5708) = 12.5·(1.5708 − 1) = 12.5·0.5708 ≈ 7.1350 square units.
Sources: Circular segment — Wikipedia
Frequently asked questions
What is the sagitta?
The sagitta is the height of the segment — the greatest perpendicular distance from the chord to the arc. It equals r·(1 − cos(θ/2)) and is useful in arch and lens design.
Why subtract sin θ from θ?
The sector area is ½·r²·θ and the triangle between the two radii and the chord has area ½·r²·sin θ. Subtracting the triangle from the sector leaves the segment, giving ½·r²·(θ − sin θ).
What happens at 180 degrees?
At 180° the chord is a diameter and the segment is exactly a semicircle, with area ½·π·r².
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This calculator is for educational purposes. Double-check important results before acting on them.