Cylinder Volume Calculator

Most people looking for the volume of a cylinder are not sitting an exam. They are standing beside a drum, a water butt, a heating oil tank or a coil of pipe, and the real question is how much it holds. This page answers the geometry and the capacity together: the cubic figure first, then litres, US gallons and Imperial gallons labelled separately, because a gallon means two different amounts depending on which side of the Atlantic you are standing on.

The cylinder volume formula

V = π r² h. Multiply pi by the radius squared, then multiply by the height. The π r² part is the area of the circular base, and multiplying by the height stacks that base all the way up the cylinder. The radius runs from the centre of the circle to its edge, the height is the perpendicular height measured straight along the axis, and pi is 3.14159.

Almost nobody measures a radius in the real world. A tape measure gives you the distance straight across, so the form you will reach for more often is V = π D² h / 4, where D is the diameter. A cylinder 10 cm across and 20 cm tall holds π × 100 × 20 / 4, which is 1,570.8 cm³. Both forms give the same answer. The second just saves you halving anything by hand.

A cubic figure is not the answer, capacity is

A volume in cubic centimetres is correct and almost useless. Divide cubic centimetres by 1,000 to get litres. Multiply cubic metres by 1,000 to get litres. One cubic metre is about 264.17 US gallons or 219.97 Imperial gallons. One cubic foot is 1,728 cubic inches, which is about 7.48 US gallons or 6.23 Imperial gallons. One US gallon is 3.78541 litres and one Imperial gallon is 4.54609 litres, a gap of roughly 20 percent, which is why nothing on this page is ever labelled with a bare word gallons.

Three capacity examples worked through

A rainwater butt 1 m in diameter and 2 m tall has a radius of 0.5 m, so V = π × 0.25 × 2 = 1.571 m³. That is about 1,571 litres, or 415 US gallons, or 345 Imperial gallons.

A tank 36 inches in diameter and 72 inches long has a radius of 18 inches, so V = π × 324 × 72 = 73,287 in³. Divide by 231 and it holds 317.26 US gallons, which is 264.2 Imperial gallons or 1,201 litres.

A water main of 300 mm bore running 1 km has a radius of 0.15 m, so V = π × 0.0225 × 1,000 = 70.7 m³, or 70,686 litres. That is the volume somebody has to drain before anyone cuts into it.

Hollow cylinders: the wall and the bore

A pipe, a tube, a sleeve or a thick walled jar is a cylinder with a smaller cylinder taken out of the middle. The wall volume, meaning the material itself, is V = π (R² - r²) h, where R is the outer radius and r is the inner radius. An outside diameter of 8 cm with a 4 cm bore over a 12 cm length gives π × (16 - 4) × 12, which is 452.4 cm³ of material.

If you know the bore and the wall thickness rather than the outside size, substitute R = r + t and the formula becomes V = π (t² + 2tr) h. The middle term is a plus, not a minus, because that is how the difference of two squares expands. Some published versions of this formula carry a minus there and they do not reproduce their own worked answers, so check any version you meet against a number you already trust.

Here is the part nobody states plainly: those same three measurements answer two completely different questions. A plumber asking how much water is inside the pipe wants the bore volume, π r² h, which for that example is 150.8 cm³. A fabricator asking how much steel is in the sleeve wants the wall volume, 452.4 cm³. Add the two together and you get the outer volume, 603.2 cm³. The hollow mode above shows all three at once and leads with the bore, because the bore is what a capacity question is really asking about.

How much water a length of pipe holds

Pipe capacity collapses into a constant once the bore is fixed, because the length is only a multiplier. Litres per metre is π d² / 4000 with the bore in millimetres, so a 15 mm bore carries 0.177 litres per metre and a 50 mm bore carries 1.963 litres per metre. Forty metres of 22 mm pipe therefore holds 40 × 0.38, which is about 15.2 litres. The lookup table above removes the arithmetic altogether. Take the bore from the pipe itself rather than from the label on the box, because nominal trade sizes and true internal bores rarely agree.

Part filled tanks, standing up and lying down

Standing on its end, a tank is easy. The liquid is simply a shorter cylinder, so V = π r² f with f as the liquid depth. Fill it to 65 percent of its height and it holds 65 percent of its capacity. Nothing surprising happens.

Lying on its side, the very same tank behaves completely differently, and this is where most estimates go wrong. The wet end of the tank is a circular segment rather than a rectangle, so the volume becomes V = L [ R² arccos((R - d) / R) - (R - d) √(2Rd - d²) ], where L is the length, R is the radius and d is the measured depth.

What that means in practice is that a dipstick misleads you at both ends of its range. A quarter of the diameter deep is only 19.6 percent of capacity. Three quarters deep is 80.4 percent. Half full really is half full, because the barrel is symmetrical, but that is true at exactly one point on the scale. The depth table above is the whole curve, and because the length and the units cancel out, the same percentages hold for a 200 litre drum and for a 30,000 litre farm tank.

Inside diameter or outside diameter

Capacity always uses the inside. It sounds obvious and it is missed constantly, because the outside of a tank is usually the only part a tape measure can reach. The error grows with the square of the diameter, so it is bigger than it feels. A tank 1 m in outside diameter with 25 mm walls has a bore of 0.95 m, and 0.95 squared divided by 1 squared is 0.9025, so it holds 90.25 percent of what the outside measurement suggests. On a nominal 1,000 litre tank that is nearly 100 litres of capacity that was never there. For a structural question, such as how much concrete a round column needs, the outside measurement is the correct one.

Five ways a cylinder calculation goes wrong

• Using the diameter where the radius belongs. This quadruples the answer, because the radius is squared. Switching the calculator above into diameter mode makes it impossible.
• Mixing units inside one calculation. A radius in centimetres with a height in metres produces a number that means nothing at all. Convert everything first, then calculate.
• Measuring a leaning cylinder along its side. An oblique cylinder still uses V = π r² h, but h has to be the perpendicular height between the two circular ends, never the length of the slanting side.
• Dropping the unit. An answer of 1,570.8 is not an answer. It is 1,570.8 cm³, and it is the cubic unit that lets you convert to litres afterwards.
• Assuming a tank on its side fills evenly. It does not, and the depth table above shows by how much.

Surface area, or the label unrolled

Peel the label off a tin and it is a rectangle, as tall as the cylinder and as wide as the circumference. That is the lateral surface area, 2 π r h, and it is the figure you want for wrapping, cladding, lagging or painting the side of something. Add the two circular ends and the total surface area is 2 π r (r + h). For a drum of 15 cm radius and 50 cm height, the side alone is 4,712 cm² and the full skin is 6,126 cm². Paint coverage is quoted per square metre, so divide by 10,000 before you read the tin.

Where cylinder volume turns up

Engine displacement is a cylinder calculation wearing overalls: the swept volume of one piston is pi times the bore radius squared times the stroke, and multiplying by the number of cylinders gives the engine size. Aquarium capacity, drum and barrel capacity, the water sitting in a borehole or a culvert, and the concrete in a round column or a pier footing all come back to the same formula.

If your vessel tapers to a point rather than running straight up, the cone volume calculator handles that case. For other solids, start at the volume calculator. For a round column of concrete there is the concrete calculator, and for the circle underneath all of this, the area calculator.