About the Circular Segment Calculator
A circular segment is the region between a chord and the arc it cuts off — the shape of an arched window, a slice of a round table with a straight edge, the liquid in a partly filled horizontal pipe seen end-on, or the bulge of a bridge arch. This circular segment calculator solves the whole segment from any of four common pairs: radius and central angle, radius and height, chord and height, or radius and chord.
It returns the segment area, chord length, arc length, the segment height (also called the sagitta or rise), the central angle, the perimeter of the segment and what share of the full circle it covers. The chord-and-height mode is especially useful in construction and woodworking, where you can measure the span and rise of an arch but not its radius — the calculator finds the radius for you.
Height can exceed the radius, in which case the segment is the major (larger) part of the circle. In radius-and-chord mode the minor segment is returned; subtract its area from πr² for the major one.
With the default inputs, the segment area is 9.0586. Change any value above to recalculate instantly.
How to use the circular segment calculator
- 1Choose which pair of measurements you know.
- 2Enter them in the same length unit (and the angle unit if needed).
- 3Read the segment area, chord, arc and height.
- 4For an arch, use chord and height to find the radius to set out the curve.
Formula and method
The segment is the sector with central angle θ minus the isosceles triangle formed by the two radii and the chord. The sector has area ½r²θ and the triangle ½r²·sin θ, giving A = ½r²(θ − sin θ) with θ in radians. For θ above 180° sin θ is negative, so the same formula adds the triangle and covers major segments too.
The chord is 2r·sin(θ/2), the arc rθ, and the height (sagitta) is the radius minus the distance from the centre to the chord, r − r·cos(θ/2). Working backwards, the intersecting chords theorem gives the radius from a span c and rise h: r = h/2 + c²/(8h); then θ = 2·arccos((r − h)/r).
- r
- Radius
- θ
- Central angle (radians in the formulas)
- c
- Chord length (span)
- h
- Segment height (sagitta, rise)
- s
- Arc length
Worked examples
Radius 10, central angle 60°
The 60° sector has area 100π/6 ≈ 52.36 and the equilateral triangle inside it ≈ 43.30, leaving a segment of about 9.06. The chord is 10 and the segment is 10(1 − cos 30°) ≈ 1.34 high.
Radius 5, height 2
cos(θ/2) = (5 − 2)/5 = 0.6, so θ ≈ 106.26°. The chord is 2 × 5 × 0.8 = 8 and the segment area is ½ × 25 × (1.8546 − 0.96) ≈ 11.18.
Arched window: 12 in span, 3 in rise
The radius is 3/2 + 144/24 = 7.5 in. The arch spans 106.26° of that circle, its curved edge is about 13.91 in long and the glass in the arch is about 25.16 in².
Major segment: chord 10, height 8
Here the rise exceeds the radius (5.5625), so the segment is the larger part of the circle: it spans about 232° and covers roughly 77% of the disc.
Frequently asked questions
What is the formula for the area of a circular segment?+
A = ½r²(θ − sin θ) with θ in radians. For r = 6 and θ = 90°, A = ½ × 36 × (1.5708 − 1) ≈ 10.27 square units.
How do I find the radius from chord length and height?+
Use r = h/2 + c²/(8h). An arch with a 4 m span and 1 m rise has radius 0.5 + 16/8 = 2.5 m.
What is the sagitta of an arc?+
The sagitta is the height of the segment — the distance from the midpoint of the chord to the arc. It equals r − √(r² − (c/2)²) for the minor segment.
What is the difference between a segment and a sector?+
A sector is bounded by two radii and an arc, like a pizza slice. A segment is bounded by a chord and an arc, so it is the sector with the triangle between the radii removed.
How do I find the area of liquid in a horizontal round tank?+
The wetted cross-section is a circular segment: enter the tank radius and the liquid depth as the height, then multiply the segment area by the tank length for the volume.