Segment Area Calculator
Enter the circle's radius and the central angle to find the area of the circular segment — the region between the chord and the arc — plus the arc length, chord length, and segment height.
°
Area of the region between the chord and the arc
9.1%
of circleSegment
9.1%
Remaining circle
90.9%
- 1
Convert angle to radians
θ = 90 × π ÷ 180 = 1.570796 - 2
Sector area = ½ × r² × θ
½ × 10² × 1.570796 = 78.5398 - 3
Triangle area = ½ × r² × sin(θ)
½ × 10² × 1 = 50The isosceles triangle formed by the two radii and the chord. - 4
Segment area = sector − triangle
78.5398 − 50 = 28.5398
How does this calculator work?
A circular segment (region between chord and arc) has area A = (r²/2)(θ − sin θ), where θ is the central angle in radians. Also: arc = r·θ, chord = 2r·sin(θ/2), sagitta = r(1 − cos(θ/2)). Enter radius and angle in degrees to get all four values plus the fraction of the circle.
Formula
How this is calculated
A circular segment is the region enclosed between a chord (a straight line connecting two points on a circle) and the arc it cuts off. It is smaller than a sector: a sector includes the two radii, a segment does not.
The area formula A = (r²/2)(θ − sin θ) comes from subtracting the triangular area (formed by the two radii and the chord) from the sector area. The sector area is (r²/2)θ and the isosceles triangle formed by the two radii has area (r²/2)sin θ, so the difference is (r²/2)(θ − sin θ). All angle calculations use radians internally — the degree input is converted by θ_rad = θ_deg × π / 180.
The sagitta (segment height) is the maximum perpendicular distance from the chord to the arc: sagitta = r(1 − cos(θ/2)). This is useful in bridge arches, lens optics, and road-curve surveying. The chord length 2r·sin(θ/2) is the straight-line distance across the opening. For a semicircle (θ = 180°) the formulas give A = πr²/2, arc = πr, and chord = 2r — consistent with standard results.
Frequently asked questions
A sector is a "pie slice" bounded by two radii and an arc. A segment is the region between a chord and the arc — it does not include the centre of the circle. For the same angle, the sector area is always larger than the segment area by the area of the triangle formed by the two radii.
The formula A = (r²/2)(θ − sin θ) requires θ in radians because sin θ and θ are in the same units only in radians. The calculator accepts degrees for convenience and converts internally using θ_rad = θ_deg × π/180.
The sagitta is the height of the segment — the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It equals r(1 − cos(θ/2)). Engineers use it to describe the "rise" of arches and bridge curves.
Also known as
TG we-Calculate Editorial Team. (2026). Segment Area Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/segment-area-calculator
TG we-Calculate Editorial Team. "Segment Area Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/segment-area-calculator.
TG we-Calculate Editorial Team, "Segment Area Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/segment-area-calculator
@misc{wecalculate_segment_area_calculator, title = {Segment Area Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/segment-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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