How to Find the Height of a Cylinder

Every version of this question is the same move: work out the area of the circle, then divide whatever you were given by it. The only thing that changes is which circle and which given. Get into the habit of naming the base area first, before touching the numbers, and the four common cases stop looking like four different problems.

Undo the base circle, not the whole formula

Volume is base area times height, so height is volume divided by base area. That is the entire rearrangement, and it works because the height only ever appears once and never gets squared. Compare that with going the other way, where the radius appears squared and needs a square root to release.

The surface area version is prettier than it sounds. Peel the label off a tin and it flattens into a rectangle, no stretching required, because a cylinder is a rolled up sheet. That rectangle is as wide as the circumference and as tall as the cylinder, so its area is 2πr times h. Divide the label area by the circumference and the height is what is left.

A worked check you can repeat on paper

The last two rows are the two wrong answers worth recognising on sight. Dividing a volume by a circumference gives 20 here, two and a half times the truth, and it happens because a length was used where an area belonged. The other one is more interesting: dividing the total surface by the circumference always gives h plus r exactly, because the two end circles contribute 2πr² and that divided by 2πr is just r. So if that route hands you 13 on a cylinder of radius 5, the answer you actually wanted was 8, and you can get there by subtracting rather than starting over.

Sizing a tank to a capacity

The practical version of this question runs backwards from a capacity. You know how many litres the thing has to hold and roughly how wide it is allowed to be, and you want to know how tall it ends up. Work in metres, since a cubic metre is 1000 litres and that keeps the numbers small.

A 1000 litre tank is 1 cubic metre. At a diameter of 1 m the base circle is 0.7854 square metres, so the tank stands 1.2732 m tall. Widen it to 1.2 m across and the base becomes 1.1310 square metres, dropping the height to 0.8842 m. That is the trade every tank designer makes: the height falls with the square of the diameter, so a small increase in width buys a large drop in height.

One caution about real containers. A 200 litre drum is 0.572 m across, so 200 litres of liquid stands 0.7783 m deep, and yet the drum itself is about 0.851 m tall. The difference is headspace and the rolled ends, not an error. Fill height and container height are different questions, and a drum that measures 0.851 m internally would hold nearer 219 litres if you filled it to the brim.

The height that uses the least metal

If the volume is fixed and you get to choose the shape, there is a best answer. Total surface area is smallest when the height equals the diameter, that is when h = 2r, and the cylinder looks square from the side. Anything taller wastes material on the curved side, anything squatter wastes it on the two ends.

Radius	Height for a volume of 628.32	Total surface area
3	22.22	475.43
4	12.50	414.69
4.64, where h is exactly 2r	9.28	406.10
5	8.00	408.41
6	5.56	435.63
8	3.13	559.20

The worked cylinder sits at 408.41 against a best possible 406.10, so it is within 0.6 percent of optimal by accident. Notice also how flat the middle of that table is and how steep the ends are: being slightly the wrong shape costs almost nothing, and being badly wrong costs a great deal.

Which raises the obvious question about drink cans, since almost none of them are shaped that way. A 330 ml can at the usual 6.6 cm across works out around 9.65 cm tall, so it stands about one and a half diameters tall rather than the single diameter the maths asks for. The optimisation above assumes the metal is the same thickness everywhere and costs the same to join, and neither is true: the ends are much thicker than the wall, and the seams cost money. Add a hand that has to hold the thing and the maths stops being the deciding vote.

Mistakes that catch people out

• Dividing a volume by the circumference. An area has to divide a volume, not a length. On the worked cylinder that slip gives 20 instead of 8, and nothing about 20 looks obviously wrong until you check it.
• Using the diameter as the radius. The base area comes out four times too big, so the height comes out a quarter of the truth. A diameter of 10 fed in as a radius turns a height of 8 into 2.
• Forgetting to remove the two ends. Total surface area includes both circles. Divide the total by the circumference and you get h plus r, which here is 13. Subtract 2πr² first, or subtract the radius at the end.
• Mixing units. A radius in centimetres with a volume in litres will not divide into anything meaningful. Put everything into one system before the first division, and remember a litre is 1000 cubic centimetres.
• Confusing fill height with container height. A 200 litre drum stands 0.851 m tall and its liquid stands 0.778 m deep. Both numbers are correct answers to different questions, and only one of them fits through a doorway.

Try these

Common questions

How do you find the height of a cylinder from its volume?

Divide the volume by the area of the base circle. A cylinder of radius 5 holding 628.32 gives 628.32 over 78.54, which is a height of 8. Work out πr² first and write it down before dividing.

How do you find the height from the surface area?

Divide the curved surface area by the circumference. If you were given the total surface area, subtract the two end circles first. A total of 408.41 on a radius of 5 leaves 251.33, and 251.33 over 31.42 is 8.

Why does dividing the total surface area by the circumference give the wrong answer?

Because it gives h plus r rather than h. The two end circles come to 2πr², and dividing that by 2πr leaves the radius sitting in your answer. Subtract the radius and you have the height.

How tall does a tank need to be to hold a given number of litres?

Turn the litres into cubic metres by dividing by 1000, then divide by the base circle in square metres. A 1000 litre tank 1.2 m across stands 0.8842 m tall, and the same tank at 1 m across stands 1.2732 m tall.

What height uses the least material for a given volume?

A height equal to the diameter, so h = 2r. For a volume of 628.32 that means a radius of 4.64 and a height of 9.28, giving a total surface area of 406.10. Real cans ignore this because the ends are thicker than the walls and the seams cost money.

Is the height the same as the depth of liquid inside?

Only when the container is full to the brim. A 200 litre drum is 0.851 m tall but the liquid stands 0.778 m deep, because the drum holds nearer 219 litres if filled completely.

Related lessons and calculators

• Volume of a cylinder, the formula this lesson divides through, with pipe capacity and part filled tanks.
• Radius of a cylinder, the harder rearrangement, since the radius is squared and has to be released with a square root.
• Height of a cone, the same question on the solid that holds a third as much, where the slant complicates matters.
• Circumference of a circle, the width of the unrolled label and the number every surface area route divides by.
• Volume of 3D shapes calculator, for cylinders, cones and frustums without the arithmetic.
• Area formulas for every common shape, the index this lesson sits inside.
• NIST on SI units, the reference for the litre and the cubic metre.