How to Find the Volume of a Cylinder

A cylinder is the shape of nearly everything that holds liquid, which is why this is one of the few volume formulas that earns its keep outside a classroom. The formula itself is two steps long. The interesting part is what happens either side of it: getting the radius right when you only have a diameter, and reading a tank that is lying on its side, where depth and contents stop agreeing entirely.

A cylinder is a prism with a round base

Every prism follows one rule: area of the end, times the distance to the other end. A box has a rectangle for an end, a triangular prism has a triangle, and a cylinder has a circle. Nothing else about the method changes, so πr²h is not a new formula so much as the old one with πr² dropped into the base slot.

You can watch that happen if you like. A regular polygon with 12 sides drawn inside a circle of radius 4 has an area of 48, so as a prism 10 tall it holds 480. Push it to 100 sides and it holds 502.32. At 10000 sides it reads 502.65, which is the cylinder. A circle is a polygon with the corners worn off, and the prism rule never noticed the difference.

A worked check you can repeat on paper

Route four is the useful one on site. You cannot always get a ruler across the middle of something, but you can nearly always get a tape around it. Measure the circumference, and the volume falls out of C²h over 4π without the radius ever being written down. A tape reading 25.13 around a post 10 tall gives 502.65, the same answer from the outside.

Standing up it is simple, on its side it is not

An upright tank is honest. Fill it to a quarter of its height and it is a quarter full, because every layer of liquid is the same circle as the one below it. Depth and contents track each other exactly, which is why a sight glass on an upright tank can be marked with a ruler.

Lay the same tank down and that stops being true. The liquid now sits in a circular segment, and near the bottom of a circle the layers are narrow. Dip a horizontal tank to a quarter of its diameter and it is 19.55 percent full, not 25. Dip it to an eighth and it is 7.21 percent full, not 12.5. Halfway is still exactly half by symmetry, and three quarters deep works out at 80.45 percent, the mirror of the quarter case.

The arithmetic for the segment is genuinely harder than πr²h and it is worth saying so. The area of liquid in the end view is r² arccos((r − d) / r) minus (r − d) times the square root of (2rd − d²), where d is the depth. Multiply that by the length of the tank. For a tank of radius 1 m and length 4 m dipped to 0.5 m, that gives 0.61418 square metres of segment and 2.4567 cubic metres of contents, against a full capacity of 12.566 cubic metres.

This is why horizontal tanks are sold with a printed dip chart rather than a plain ruler. If someone hands you a dipstick reading and asks for litres, do not scale it linearly. That mistake runs about 28 percent high at quarter depth, which on a 12566 litre tank is 685 litres of contents that are not there.

Litres, and the number plumbers keep in their head

A litre is 1000 cubic centimetres and a cubic metre is 1000 litres, so the only real decision is which unit to measure in before you start. Work in metres and the answer arrives in cubic metres. Work in centimetres and it arrives in millilitres, since a millilitre and a cubic centimetre are the same thing.

Pipe capacity is the version of this calculation that gets used most often, because it sets how much water sits in a system and how long it takes to run hot. A 100 mm bore holds 7.854 litres per metre. Anything else scales by the square of the bore, so doubling the pipe size holds four times as much water per metre and not twice.

Cylinder	Radius	Height or length	Volume
Worked example	4	10	502.65
15 mm bore pipe, per metre	7.5 mm	1 m	0.177 litres
22 mm bore pipe, per metre	11 mm	1 m	0.380 litres
100 mm bore pipe, per metre	50 mm	1 m	7.854 litres
Water tank, 1.2 m across	0.6 m	1.5 m	1696.5 litres
Concrete pier, 300 mm across	0.15 m	2.4 m	0.170 cubic metres

The cone and the sphere hiding inside it

The cylinder is the yardstick the other round solids are measured against. A cone standing on the same circle with the same height holds exactly one third: 167.55 against 502.65. Archimedes went further and showed that a sphere fits its own snug cylinder, the one with a height equal to the diameter, at exactly two thirds. For a radius of 4 that cylinder holds 402.12 and the sphere inside it holds 268.08.

Put those together on the same base and height and the cone, the sphere and the cylinder come out as 1 to 2 to 3. Archimedes asked for the sphere and cylinder to be carved on his tombstone, which tells you what he made of it. It is also a fast way to sanity check a cone or sphere calculation: work out the cylinder, which is easy, then take a third or two thirds.

Mistakes that catch people out

• Using the diameter where the radius belongs. This is the expensive one, because the error is squared. A pipe of 100 mm treated as radius 100 comes out four times too big, and the answer still looks perfectly reasonable on the page.
• Squaring the wrong letter. πr²h and πrh² are not the same quantity. Only the radius is squared, because only the radius appears twice in the circle.
• Trusting the nominal pipe size. Nominal is not bore. A 15 mm copper pipe has an internal diameter nearer 13.6 mm, so it really carries about 0.145 litres per metre rather than 0.177, roughly 18 percent less than the label suggests.
• Reading a horizontal tank as though it were upright. A dipstick is a depth, not a percentage. On a tank lying down, a quarter of the depth is 19.55 percent of the contents.
• Assuming the object is a plain cylinder. Drink cans, buckets and plant pots all taper, so their layers are not identical and πr²h will not describe them. A tapered container needs the frustum rule instead.

Try these

Common questions

What is the formula for the volume of a cylinder?

V = πr²h. Square the radius, multiply by π to get the area of the end, then multiply by the height. A cylinder of radius 4 and height 10 holds 502.65 cubic units, which is exactly 160π.

What if I only know the diameter?

Halve it first, then carry on as normal. A diameter of 10 with a height of 4 gives π times 25 times 4, which is 314.16. Feeding the diameter straight into the formula gives 1256.64, four times too much, because the mistake gets squared.

How do I find the volume from the circumference?

Use C²h divided by 4π. A tape reading 25.13 around a cylinder 10 tall gives 502.65 without ever working out the radius, which is handy when you can get a tape around something but not a ruler across it.

How many litres does a cylinder hold?

Work in metres and multiply the answer in cubic metres by 1000, or work in centimetres and divide by 1000. A tank of radius 0.6 m and height 1.5 m holds 1.6965 cubic metres, which is 1696.5 litres.

Why is a tank on its side not half full at half depth?

At exactly half depth it is half full, by symmetry. Everywhere else it is not, because the liquid sits in a circular segment and the layers near the bottom are narrow. A quarter of the depth is 19.55 percent of the contents, and an eighth is 7.21 percent.

How much water is in a metre of pipe?

A 100 mm bore holds 7.854 litres per metre, a 50 mm bore holds 1.9635 and a 22 mm bore holds 0.380. Capacity follows the square of the bore, so halving the pipe size quarters the water it carries.

Related lessons and calculators

• Radius of a cylinder, the same formula run backwards when the volume is the thing you already know.
• Volume of a cone, exactly one third of the cylinder it stands in.
• Volume of a sphere, exactly two thirds of its own snug cylinder.
• Volume of a rectangular prism, the same base area times height rule with a rectangle in the base slot.
• Area of a sector, where the segment maths behind a horizontal tank comes from.
• Volume of 3D shapes calculator, including the frustum rule for anything that tapers.
• Area formulas for every common shape, the index this lesson sits inside.
• BIPM on measurement units, the authority behind the cubic metre and the litre.
• Height of a cone, worked back from a volume, a slant, a surface area or an angle.
• Height of a cylinder, the same formula divided through when the volume is what you already have.