Circular segments appear in tank volumes (a cylinder lying on its side is partly full), arched windows, bridge arches and the shape of a lens. Their area is the sector minus the triangle under the chord.
How to use the Segment Area Calculator
- Radius r
- Also known
- Central angle α
- Segment height h
Results update as you type, and everything is calculated in your browser.
How it works
Enter the radius and the angle or height. If h is greater than r the segment is the larger part of the circle, which the formula handles automatically.
Inputs explained
| Field | What to enter |
|---|---|
| Radius r | Units: cm, m, mm, in, ft, yd … |
| Also known | Choose one: Central angle α, Segment height h |
| Central angle α | Units: °, rad, π rad, grad, turn |
| Segment height h | Units: cm, m, mm, in, ft, yd … |
What each result means
- Segment area = [Radius r]^2×(rad([Central angle]) – sin(rad([Central angle])))÷2
- Arc length = [Radius r]×rad([Central angle])
- Chord length = 2×[Radius r]×sin(rad([Central angle])÷2)
- Share of the circle = [Radius r]^2×(rad([Central angle]) – sin(rad([Central angle])))÷2÷(pi×[Radius r]^2)×100
Example values
How the main result changes when Radius r changes and everything else stays at the example values above:
| Radius r (cm) | Segment area (cm²) |
|---|---|
| 5 | 7.135 |
| 7.5 | 16.0536 |
| 10 | 28.5398 |
| 12.5 | 44.5935 |
| 15 | 64.2146 |
| 20 | 114.1593 |
Formula
A = r²(α − sin α)/2, α = 2·arccos((r − h)/r)
Worked example
Inputs — Radius r: 10 cm; Also known: Central angle α; Central angle α: 90 °.
Results — Central angle = 90 °; Segment area = 28.539816 cm²; Arc length = 15.707963 cm; Chord length = 14.142136 cm; Share of the circle = 9.0845 %.
Key terms
- Circular segment
- The area between a chord and its arc.
Frequently asked questions
What is a circular segment?
The region cut off by a chord.
How is it different from a sector?
A sector is bounded by two radii; a segment by a chord.
How do I find the liquid in a horizontal tank?
Segment area × tank length.
What if h = r?
The segment is a half-circle.