Circular Segment Area Calculator

A circular segment is the region between a chord, meaning a straight line cutting across the circle, and the arc that chord cuts off. Look at a round glass lying on its side with a little water in it: the water’s flat surface is the chord, and the crescent of water beneath it is the 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 tank has a 1 m radius, so it is 2 m across. The water is 30 cm deep at the bottom. What is the cross-sectional area of the water?

The water surface is the chord. With depth h = 0.3 m and radius r = 1 m, the central angle of 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.

The full circle is π = 3.1416 m², so the tank is 9.4% full. The water stands 15% of the way up the tank and holds barely nine percent of it.

Why this trips people up

The instinct is to read the dipstick as a percentage: water 15% of the way up means the tank is 15% full. It never is, on a cylinder lying on its side. The bottom of the tank is a narrow wedge, so the first few centimetres of depth cover very little width and hold very little liquid. Around the middle the tank is at its widest and each centimetre is worth far more.

The relationship is not linear at all, and only crosses at the halfway mark: half depth really is half full, by symmetry. Below that the true fill runs under the depth reading, and above it runs over. At 15% depth you have 9.4%; at 85% depth you have 90.6%.

Where segments show up:

• Horizontal cylindrical tank fill. Heating oil, propane and water tanks lying on their side. Turning a dipstick reading into litres is exactly this calculation.
• 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. ✓