How to Find the Radius of a Cylinder

The volume formula is built forwards: take a circle, carry it up through a height. Working back to the radius means reversing both of those steps, in the opposite order to the one they were applied in. Get that order wrong and the arithmetic still produces a number, just not the right one.

Undo the height first, then the square

V = πr²h has three things done to the radius: it is squared, multiplied by π, then multiplied by the height. Reversing that means dividing by the height, dividing by π, and only then taking the square root. Doing the root first, on the raw volume, gives 25.07 for the example above rather than 5, which is not close to anything useful.

A cleaner way to hold it in your head: dividing the volume by the height gives you the area of the base circle, and finding a radius from a circle’s area is a problem you already know. For the worked example that is 628.32 over 8, which is 78.54, and the radius of a circle with that area is 5.

The rearrangement, drawn out

Written as one line it is r = √(V / πh), but the line hides the order of operations that matters. Set out as three separate steps it is much harder to get wrong, and each intermediate number means something you can sanity check.

A worked check you can repeat on paper

Route four is the one worth remembering for real objects. The label wrapped round a tin is the lateral surface, 2πrh, so if you can measure the label and you know the height, the radius falls straight out without opening anything or knowing the volume at all.

When you are given the surface area instead

This case is genuinely harder, and it is worth saying so plainly rather than pretending it is another rearrangement. Total surface area is 2πr² + 2πrh, and the radius appears both squared and on its own, so there is no way to isolate it. What you have is a quadratic.

Take the same cylinder, whose total surface area is 408.41, with a height of 8. Divide through by 2π and it tidies to r² + 8r = 65, so r² + 8r − 65 = 0. That factors as (r + 13)(r − 5) = 0, giving r = 5 or r = −13. Discard the negative one, because a radius cannot be negative, and you are left with 5. The tidy factors are a consequence of the numbers chosen here; in general you would reach for the quadratic formula and keep the positive root.

Why the radius matters more than the height

The radius is squared in the volume formula and the height is not, which makes them very unequal partners. Double the height and a cylinder holds twice as much. Double the radius and it holds four times as much. Halve the radius and three quarters of the contents disappear.

Change to the worked cylinder	New volume	Factor
Nothing, radius 5 and height 8	628.32	1
Height doubled to 16	1256.64	2
Radius doubled to 10	2513.27	4
Radius halved to 2.5	157.08	0.25

It also means a small measuring error in the radius hurts twice as much as the same error in the height. If you are working back to a radius in order to check a manufactured part, that is the number to measure carefully.

One practical note on units. A millilitre is exactly one cubic centimetre, so a can holding 330 ml with 11.5 cm of internal height needs a radius of √(330 / (π × 11.5)), which is 3.02 cm, so about 6.04 cm across. Real 330 ml cans measure closer to 6.6 cm, because the body tapers at both ends and there is headspace above the liquid. The arithmetic is not wrong, the can simply is not a plain cylinder.

Mistakes that catch people out

• Taking the square root too early. The root goes last, after both divisions. Rooting the volume itself gives 25.07 here instead of 5.
• Dividing by 2π out of habit. That is the circumference rearrangement. The volume route divides by π once, not by 2π.
• Reporting the diameter as the radius. This one is quiet, because 10 looks as plausible as 5. If the answer feeds back into a volume it will come out four times too big, which is the tell.
• Mixing litres with centimetres. Convert first. A litre is 1000 cubic centimetres, and a millilitre is exactly one cubic centimetre.
• Trying to isolate r from the surface area. It cannot be done by rearranging, because r appears squared and linear in the same expression. Set it up as a quadratic and solve it properly.

Try these

Common questions

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

Divide the volume by π and by the height, then take the square root. A volume of 628.32 with a height of 8 gives 25, and the square root of 25 is a radius of 5.

Can you find the radius without the height?

Only if you have something that belongs to the base circle by itself, such as its circumference or its area. From the volume alone the answer is no, because a tall thin cylinder and a short wide one can hold exactly the same amount.

How do you find the radius from the circumference?

Divide by 2π. A base circumference of 31.42 gives a radius of 5. This is often the easiest measurement to take on a real object, since a tape round the outside is simpler than reaching across the middle.

How do you find the radius from the surface area?

You solve a quadratic, because the radius appears both squared and on its own in 2πr² + 2πrh. Divide through by 2π first, which turns a surface area of 408.41 with a height of 8 into r² + 8r − 65 = 0, and the positive root is 5.

Why does the square root come last?

Because the squaring happened first when the formula was built. Reversing a chain of operations means undoing them in the opposite order, so the multiplications by π and the height come off before the square does.

Does doubling the radius double the volume?

No, it quadruples it, because the radius is squared. Doubling the height is what doubles the volume. That difference is why a small error in a measured radius matters roughly twice as much as the same error in a height.

Related lessons and calculators

• Circumference of a circle, the quickest route to a radius when you can get a tape round the outside.
• Surface area of a cone, another solid where the measurement you are given is rarely the one the formula wants.
• Surface area of a rectangular prism, for the difference between covering a solid and filling it.
• Area formulas for every common shape, including the circle this whole calculation rests on.
• Volume of 3D shapes calculator, for running cylinders and cones forwards without the arithmetic.
• NIST on SI units, the reference for why a millilitre and a cubic centimetre are the same thing.
• Volume of a pyramid, for the solid that holds a third of the box rather than filling it.
• Volume of a sphere, the solid that fills exactly two thirds of the cylinder it fits inside.
• Volume of a cylinder, the forward calculation this lesson unwinds, with dip readings and pipe capacity worked out.
• Height of a cone, the same rearranging problem on the solid a cylinder holds three of.
• Height of a cylinder, the easier of the two rearrangements, since the height is never squared.