Circle Segment Area Calculator
Find the area of a circular segment from the radius plus a central angle in degrees or radians — or from the segment height (sagitta) if that's what you can measure. Chord, arc length, and angle come with it.
Example: with Radius (r) 10 · Second measurement Central angle in degrees · Angle or height value 90 → Segment area: 28.54 square units.
- Chord length14.142 units
- Arc length15.708 units
- Segment height2.929 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
A = ½r²(θ − sin θ): sector area minus the triangle under the chord, with θ in radians.
Sector minus triangle
A segment is the region between a chord and the arc above it. Slice out the full sector (the pizza slice, area ½r²θ) and remove the triangle formed by the two radii and the chord (area ½r² sin θ): what remains is the segment, A = ½r²(θ − sin θ). The angle must be in radians for the subtraction to make sense — degree inputs are converted first.
Past 180° the sine turns negative, so the formula automatically adds the triangle back and keeps working all the way to a full circle.
When all you can measure is the height
In the field you often cannot see the center — a partly filled horizontal tank, a stone arch, a stadium cross-section. What you can measure is the chord and the bulge above it: the sagitta h. Given r and h, the angle is recovered with θ = 2·arccos(1 − h/r), and everything else follows. That is exactly what the segment-height mode does.
How it’s calculated
A = ½r²(θ − sin θ) with θ in radians. Degree mode converts by π/180; height mode recovers the angle with θ = 2·arccos(1 − h/r). Chord = 2r·sin(θ/2); arc = rθ; height = r(1 − cos(θ/2)). Angles accepted up to 360° (2π); rounded to 3 decimals.
Computes a single circular segment; for the area between two parallel chords, run the tool twice and subtract.
Segment area grows fast with angle (r = 10)
| Central angle | Segment area |
|---|---|
| 30° | 1.18 |
| 60° | 9.059 |
| 90° | 28.54 |
| 120° | 61.418 |
| 180° | 157.08 (half circle) |
Computed with A = ½r²(θ − sin θ) for r = 10; rounded to 3 decimals.
Common mistakes
- Using degrees directly in ½r²(θ − sin θ) — θ must be in radians, or use this page's degree mode.
- Confusing the segment (between chord and arc) with the sector (the full pizza slice including the triangle).
- Treating the chord as the arc length; the arc rθ is always longer than the chord.
- Entering a segment height greater than the diameter 2r — geometrically impossible; re-measure.
Frequently asked questions
What is the formula for the area of a circular segment?
A = ½r²(θ − sin θ), where r is the radius and θ the central angle in radians. It is the sector area ½r²θ minus the triangle ½r² sin θ.
What is the difference between a segment and a sector?
A sector is the wedge from the center out to the arc, like a pizza slice. A segment is only the part between the chord and the arc — the sector with its central triangle cut away.
I only know the chord and the height. How do I get the area?
First recover the radius: r = h/2 + c²/(8h), where c is the chord and h the height. Then switch this page to segment-height mode with that r and h.
Can the central angle be more than 180°?
Yes — that is a major segment, the bigger piece of the circle. The same formula handles it because sin θ goes negative, and at 360° you get the full circle area πr².