Why a cylinder is just an area with a length
A cylinder has the same circular cross-section from one end to the other, so its volume is nothing more than that cross-sectional area multiplied by the length. Work out the area of the circle, πr², then stretch it along the axis by h. That gives V = πr²h, and there is no correction factor of any kind because nothing tapers.
Cavalieri's principle extends the result further than you might expect. If you shear a stack of coins sideways so the pile leans, the volume does not change: every horizontal slice still has the same area, and the height from bottom to top is unchanged. So an oblique cylinder holds exactly as much as an upright one of the same perpendicular height. What does change is the surface area, since the curved surface of a leaning cylinder is longer.
Because the radius is squared, cross-sectional area is the sensitive parameter. Doubling the diameter quadruples the capacity, which is why a modest increase in bore transforms a pipe's carrying volume. A 2-inch pipe holds four times as much per foot as a 1-inch pipe, and a 6-inch pipe holds nine times as much as a 2-inch one.
Capacity per foot: the number the trades actually use
For pipe runs, well casings and tank dipping, the useful figure is not the total volume but the volume per unit of length. It is the base area converted straight into the trade unit: gallons per foot = πr² × 7.480519, with r in feet. Once you know it, you multiply by any length or any depth and you are done.
The standard plumbing figures come straight out of this. A 4-inch bore is a radius of 1/6 foot, so the base area is π/36 = 0.087266 ft² and the capacity per foot is 0.087266 × 7.480519 = 0.6528 gallons per foot. A 2-inch bore is a quarter of that at 0.1632 gal/ft; an 8-inch bore is four times it at 2.611 gal/ft. This calculator reports the same figure for whatever bore you enter.
The important caveat is that nominal pipe size is not internal diameter. A pipe described as 4-inch nominal has an outside diameter of 4.5 inches and an internal bore that depends on the wall schedule — 4.026 inches in Schedule 40 steel. Using 4 inches instead of 4.026 understates the capacity by 1.3%; using the 4.5-inch outside diameter overstates it by 25%. Always work from the bore.
Worked example: a 6-foot-diameter water tank
An upright cylindrical tank has an internal diameter of 6 feet and stands 8 feet tall. Find its capacity in gallons and the gallons per foot of depth.
- Halve the diameter. r = 6 ÷ 2 = 3 ft.
- Find the base area. π × 3² = π × 9 = 28.2743 ft².
- Multiply by the height. 28.2743 × 8 = 226.195 ft³.
- Convert to gallons. 226.195 × 7.480519 = 1,692.06 US gallons.
- Find gallons per foot of depth. 28.2743 × 7.480519 = 211.51 gal/ft. Check: 211.51 × 8 = 1,692.1 gallons, which matches.
- Find the material area. Curved surface 2π × 3 × 8 = 150.80 ft², plus two ends at 28.2743 each, giving 207.35 ft² in total.
The gallons-per-foot figure is what makes a dipstick useful. Measure 5 feet 3 inches of water in this tank and the contents are 5.25 × 211.51 = 1,110.4 gallons. That linear relationship holds only because the tank stands on end; lay the same tank on its side and the contents against depth follow a curve, not a straight line.
Upright versus lying down, and other cautions
The dip table produced here assumes the cylinder stands on end, so contents rise in exact proportion to depth: half the depth is half the volume. That is the easy case and it is why vertical tanks are so much simpler to gauge.
A cylinder lying on its side behaves completely differently. At half depth it is exactly half full by symmetry, but at a quarter depth it holds only about 19.6% of its capacity, because the cross-section at the bottom is a narrow circular segment. Anywhere other than the exact midpoint, treating a horizontal tank as linear will mislead you, and you need a segment-area calculation instead.
Two more cautions. First, a real tank is rarely a bare cylinder: dished or domed ends add volume beyond the πr²h figure, and internal fittings subtract it. Second, quoted tank capacity is usually working capacity rather than geometric capacity, since fill lines, ullage and expansion space are deducted. A drum with a 22.5-inch bore and 33.5 inches of internal height computes to 57.7 gallons of geometric volume but is sold as a 55-gallon drum for exactly that reason.
Capacity per foot for common bores
| Internal diameter | Base area (ft²) | US gal per ft | Litres per metre |
|---|---|---|---|
| 1 in | 0.005454 | 0.0408 | 0.5067 |
| 2 in | 0.021817 | 0.1632 | 2.0268 |
| 3 in | 0.049087 | 0.3672 | 4.5604 |
| 4 in | 0.087266 | 0.6528 | 8.1073 |
| 6 in | 0.196350 | 1.4688 | 18.2415 |
| 8 in | 0.349066 | 2.6112 | 32.4293 |
| 12 in | 0.785398 | 5.8752 | 72.9659 |
| 3 ft | 7.068583 | 52.877 | 656.69 |
| 6 ft | 28.27433 | 211.51 | 2,626.77 |
Litres per metre is the same quantity in metric: multiply the base area in square metres by 1,000. A 4-inch bore is 8.107 litres per metre of run.
Mistakes that ruin a capacity estimate
- Using the diameter as the radius. Because the radius is squared, this overstates the volume by a factor of four. It is the single most common error on this shape.
- Using nominal pipe size as the bore. A 4-inch Schedule 40 steel pipe has a 4.026-inch internal diameter and a 4.5-inch outside diameter. Only the bore carries fluid.
- Treating a horizontal tank as linear. A cylinder on its side is half full at half depth but holds only about 19.6% at quarter depth. Use a segment calculation instead.
- Forgetting dished ends. Domed heads on a pressure vessel add real volume that πr²h does not include; the manufacturer's data sheet is the authority.
- Mixing gallon definitions. A US gallon is 231 cubic inches; an imperial gallon is about 20% larger. This calculator uses the US gallon throughout.
A fast mental estimate for round tanks
For an upright tank measured in feet, gallons ≈ 5.875 × d² × h, with d the diameter in feet. The constant is π ÷ 4 × 7.480519 = 5.8752. For a 6-foot tank 8 feet tall that gives 5.875 × 36 × 8 = 1,692 gallons, matching the exact calculation. In metric the equivalent shortcut is litres ≈ 785.4 × d² × h with both dimensions in metres.
Related shapes and where they take over
Once a shape tapers, the cylinder formula stops applying. A cone holds exactly one third as much as the cylinder that contains it, so a conical hopper bottom needs the cone volume calculator, and a tapered silo transition needs the frustum calculator. A domed tank head is a spherical cap, handled by the spherical cap volume calculator.
For a tank that is neither round nor tapered, use the rectangular prism calculator. And where you only need the flat cross-section — to size a pump inlet or check a flow velocity — the circle area calculator gives you that directly from the bore.
Surface area alone, for insulation, cladding or painting, is often the quantity that costs money rather than the volume. The cylinder surface area calculator separates the curved area from the ends, which matters because the two are usually treated differently in a coating specification.