Circular Segment Area Calculator
Compute the area of a circular segment (the region between a chord and the arc it cuts off).
Includes horizontal-tank fill scenarios.
A circular segment is the region between a chord (a straight line cutting through the circle) and the arc the chord cuts off. Picture the moon in a half-full glass — that crescent of water is a segment.
The cleanest formula uses the central angle θ (the angle at the center, between the two radii drawn to the chord’s endpoints):
A = ½ × r² × (θ − sin θ)
θ must be in radians. The (θ − sin θ) factor is the difference between the sector area and the triangle cut out by the chord and the two radii.
Worked example — water in a horizontal cylindrical tank: A horizontal cylindrical water tank has a 1 m radius. The water is 30 cm deep at the bottom. What’s the cross-sectional area of the water?
The water surface is the chord. With water depth h = 0.3 m and radius r = 1 m, the central angle θ for the wet segment is θ = 2 × arccos((r − h) / r) = 2 × arccos(0.7) ≈ 1.5908 rad. A = 0.5 × 1² × (1.5908 − sin(1.5908)) = 0.5 × (1.5908 − 0.99996) ≈ 0.2954 m².
So the water cross-section is about 0.295 m² of the total 3.14 m² circle. The tank is 9.4% full — much less than the “30% of the diameter” depth would suggest, because the bottom of a horizontal cylinder is narrow.
Why this trips people up:
A common gut estimate says “the tank is 30% full because the water is 30% as deep as the diameter.” That’s wrong for horizontal tanks. The bottom holds less per inch of depth than the middle does. Real fill is much lower at the bottom and much higher in the upper-middle.
Where segments show up:
- Horizontal cylindrical tank fill. Heating oil, propane, water tanks lying on their side — the depth-to-volume relationship is a segment problem.
- Arched windows above a door. The half-moon glass panel area for stained-glass or paint estimating.
- Tunnel cross-sections. Drainage culverts, half-pipe tunnels.
- Partially-buried pipes. Estimating soil load on a pipe sitting in a trench.
The other formula form uses chord length c and segment height (sagitta) h: A = r² × arccos((r − h)/r) − (r − h) × √(2rh − h²) Same answer, different inputs. Useful when you measure the segment directly rather than via central angle.
Sanity check:
- θ = 0: A = 0 (no segment). ✓
- θ = 2π (full revolution): A = π × r² (the whole circle). ✓
- θ = π (half circle, chord = diameter): A = ½πr², a semicircle. ✓