Formula
A vertical cylinder is π × radius² × the fill, and the fill cannot exceed the shell height. A rectangle is length × width × fill. A horizontal cylinder uses the circular segment. With h measured up from the bottom and R the radius, the area is R² arccos((R − h) / R) − (R − h) √(2 R h − h²). Volume is that area times the length. At h = 0 the area is 0. At h = R, arccos(0) is π/2, and the area is half the circle. At h = the diameter, the area is the whole circle.
half of a horizontal tank = ½ × π × R² × length
gallons = cubic feet × 7.48052
The dipstick is the same segment at every tenth of the diameter. Ends are square to the barrel. A dished head is not included. The line that leaves the tank is the pipe volume calculator and the friction loss calculator.
Worked examples
Horizontal, 4 ft diameter, 10 ft long, filled to 2 ft. That is the midline. Volume is half of π × 2² × 10 = 62.832 ft³, about 470 gallons.
The same tank filled to 1 ft. The segment is 4 × π/3 − √3 = 2.457 ft², times 10 ft is 24.57 ft³.
A vertical tank of the same shell, full. π × 2² × 10 = 125.66 ft³. The horizontal full tank is the same number. Only the partial fill differs.
Reference
Dipstick for the 4 ft × 10 ft horizontal tank in the first example. Other sizes recompute on the page.
| Fill | Depth, ft | ft³ | Gallons |
|---|---|---|---|
| 0% | 0 | 0 | 0 |
| 10% | 0.4 | 6.54 | 49 |
| 20% | 0.8 | 17.89 | 134 |
| 30% | 1.2 | 31.71 | 237 |
| 40% | 1.6 | 46.94 | 351 |
| 50% | 2 | 62.83 | 470 |
| 60% | 2.4 | 78.72 | 589 |
| 70% | 2.8 | 93.96 | 703 |
| 80% | 3.2 | 107.77 | 806 |
| 90% | 3.6 | 119.12 | 891 |
| 100% | 4 | 125.66 | 940 |
Questions
What is the horizontal-tank formula?
The filled part of the circle is a circular segment. Area = R² arccos((R − h) / R) − (R − h) √(2 R h − h²), and the volume is that area times the tank length. Half full is half the cylinder, which is a useful check.
Is the dipstick the same formula?
Yes. Each row is the segment at that fraction of the diameter. A 4 ft tank at half depth is the 50% row.
Does this include the dish ends?
No. Both cylinders are straight barrels. A dished head holds more than this. Measure the straight side and add the heads from the manufacturer’s drawing.
Sources
- Circular segment: A = R² arccos((R − h) / R) − (R − h) √(2 R h − h²). Half full reduces to half the cylinder.
- 1 ft³ = 7.48052 US gallons. Straight barrel only. Dished heads are not included.