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CalcMax

Cylinder Volume Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

113.10 cm³

Volume

Base area
28.27 cm²
Lateral area
75.40 cm²
Total surface area
131.95 cm²

A cylinder calculator takes the radius of the circular end and the height between the two ends and returns four numbers: the volume inside it, the area of one end, the area of the curved side, and the total surface area of the whole thing including both ends. A tin of beans, a water tank, a concrete pillar and a roll of tape are all cylinders, and the two measurements you need are the two you can take with a tape measure — across the top and up the side. The arithmetic is two formulas. The volume is the area of the circular end multiplied by the height, which is π times the radius squared times the height. The total surface area is the two ends plus the curved side: each end is π times the radius squared, and the side unrolled is a rectangle whose width is the circumference of the end and whose height is the height of the cylinder, so it comes to two π times the radius times the height. Adding them gives two π times the radius times the radius plus the height. The page shows the two pieces separately as well as the total, which matters more than it might sound: a question about how much paint a tank needs is a question about the side alone, and a question about how much metal the ends need is a question about the base area alone, and a page that only gave the total would make you take it apart yourself. Two of the four outputs are worth a second look before you trust them. They are in cubic and square centimetres whatever unit the dropdown beside the inputs is set to — that is a deliberate choice rather than an oversight, since the four outputs are a fixed set and the reference table below is computed in the same unit, so switching the input to inches changes how your number is read on the way in and nothing about how the answer is written. If you measured in inches, convert the answer back yourself: divide the volume by 16.387 and the areas by 6.4516. And the surface area is computed from the unrounded radius rather than from the two areas printed above it, so adding the printed base area to the printed side area can be a hundredth out. The printed total is the correct one; the discrepancy is what happens when a number that never ends gets written down twice.

Common cylinders and what they come to

Radius (cm)Height (cm)Volume (cm³)Base area (cm²)Lateral area (cm²)Surface area (cm²)
512942.4878.54376.99534.07
34113.128.2775.4131.95
68904.78113.1301.59527.79
10103141.59314.16628.321256.64
2675.412.5775.4100.53
7.5203534.29176.71942.481295.91
113.143.146.2812.57
126.283.1412.5718.85

Eight cylinders, and the first row is the one the page loads with. Read the second and third rows together: doubling the radius from 5 to 10 and shrinking the height from 12 to 8 multiplies the volume by about ten, which is the square in πr² at work and is why a wide tank beats a tall one when the footprint is not the constraint. The fourth row is the shape everyone draws, a cylinder as tall as it is wide, and its lateral area is exactly twice its base area — a relation that holds whenever the height equals the radius rather than anything special about tens. The fifth row carries the coincidence worth meeting before you write in about it: at a radius of 2 the volume and the lateral area are both 75.4, because πr²h equals 2πrh only when r is 2, for every height. The sixth row is the same kind of coincidence on a different pair of columns — at a height of 1 the volume and the base area agree, since multiplying by one changes nothing — and the seventh row is the unit cylinder where both hold at once. The last row is a tall thin one, a radius of 7.5 and a height of 20, giving a little over three and a half litres. Every figure here is recomputed from its radius and height when the page is built, in centimetres, and the two decimals are the same two decimals the results panel uses.

Formula

V = πr²h A_base = πr² A_side = 2πrh A_total = 2πr² + 2πrh = 2πr(r + h)

Radius
The distance from the centre of an end to its edge, in centimetres. This is half the diameter, so a cylinder quoted across the top rather than across the middle needs dividing by two first. Zero is allowed and gives a cylinder with no width, in which case all four outputs are zero
Height
The perpendicular distance between the two circular ends, in centimetres. It is not the distance across the curved side from one rim to the opposite rim — that is longer, since it goes around rather than straight up, and using it would overstate the volume
Volume
The space inside the cylinder, in cubic centimetres: the area of one end multiplied by the height. A cubic centimetre is a millilitre, so a volume in these units converts straight into litres by dividing by a thousand
Base area
The area of one circular end, in square centimetres, π times the radius squared. Both ends are the same size, so this is the area of either of them rather than of the two together
Lateral area
The area of the curved side alone, in square centimetres. Unrolled, that side is a rectangle of height h and width 2πr, which is where the formula comes from and is the easiest way to see why the height and the circumference both appear in it
Total surface area
The two ends plus the curved side, in square centimetres. Closely related to the volume but not proportional to it: doubling the height doubles both, while doubling the radius multiplies the volume by four and the total surface area by less than four
π
The ratio of a circle's circumference to its diameter, about 3.14159. The page uses the full double-precision value rather than shortening it, so a hand calculation done with 3.14 will drift from the printed answer in the last place
Two decimal places
How wide every output is written. Two rather than four because these are physical measurements of real objects — a radius taken off a ruler is itself uncertain by more than the third decimal, so printing more digits would suggest a precision the input never had

The page answers two different kinds of question with the same two inputs. The first is capacity: how much a tank, a drum, a pipe or a tin holds. That is the volume, and it is worth knowing that a cylinder's volume grows with the square of its radius, so a tank twice as wide holds four times as much while one twice as tall holds only twice as much — which is why wide tanks beat tall ones when space is short. The second is material: how much sheet metal a can needs, how much paint a pillar takes, how much wrapping a roll wants. Those are the surface areas, and the page separates the ends from the side because those are usually supplied differently — sheet metal is cut from one stock, the labels or the paint go on the side. The lateral area is also the one that surprises people, because unrolling the side into a rectangle is not obvious until you have done it once: a cylinder of radius 5 and height 12 has a side that unrolls into a rectangle 31.4 centimetres wide and 12 tall. One conversion is worth carrying around: a volume in cubic centimetres is a volume in millilitres, so a cylinder of 942.48 cubic centimetres holds 0.942 litres, and the step is a division by a thousand rather than anything more interesting.

Worked examples

  1. A radius of 3 and a height of 4

    1. Base area: π × 3 × 3 = 28.274333…, which rounds to 28.27
    2. Volume: 28.274333… × 4 = 113.097335…, which rounds to 113.1
    3. Lateral area: 2 × π × 3 × 4 = 75.398223…, which rounds to 75.4
    4. Total surface area: 2 × 28.274333… + 75.398223… = 131.946891…, which rounds to 131.95

    The input the page loads with. The volume of 113.1 cubic centimetres is the same figure the cone page returns for the same two numbers, which is not a coincidence: a cone is a third of the cylinder it fits inside, and that page's 113.1 is its surface area rather than its volume. Reading the two pages side by side is worth doing once so the match does not look like a mistake. The check that needs no calculator: a radius of 3 gives an end of about 28 square centimetres, four of those stacked is about 113.

  2. A radius of 5 and a height of 12

    1. Base area: π × 5 × 5 = 78.539816…, which rounds to 78.54
    2. Volume: 78.539816… × 12 = 942.477796…, which rounds to 942.48
    3. Lateral area: 2 × π × 5 × 12 = 376.991118…, which rounds to 376.99
    4. Total surface area: 2 × 78.539816… + 376.991118… = 534.070751…, which rounds to 534.07

    The one to keep if you only keep one, because the side unrolls into a rectangle you can picture: 2π × 5 is 31.4159 centimetres around, and a rectangle 31.4159 wide by 12 tall is 376.99 square centimetres, which is the lateral area. The volume in millilitres is 942.48, so this cylinder holds a little under a litre. Doubling the height to 24 would double the volume and the lateral area but leave the base area at 78.54, which is the cleanest way to see which outputs scale with which input.

  3. A radius of 10 and a height of 10

    1. Base area: π × 10 × 10 = 314.159265…, which rounds to 314.16
    2. Volume: 314.159265… × 10 = 3141.592654…, which rounds to 3141.59
    3. Lateral area: 2 × π × 10 × 10 = 628.318531…, which rounds to 628.32
    4. Total surface area: 2 × 314.159265… + 628.318531… = 1256.637061…, which rounds to 1256.64

    A cylinder as tall as it is wide, which is the shape most people draw when they are asked to draw one. The volume is a thousand times π, and because a cubic centimetre is a millilitre that is 3.14 litres. Notice that the lateral area is exactly twice the base area here, and that this is a consequence of h being equal to r rather than anything special about tens: the same doubling happens at any radius and height that match.

  4. A radius of 2 and a height of 6

    1. Base area: π × 2 × 2 = 12.566370…, which rounds to 12.57
    2. Volume: 12.566370… × 6 = 75.398223…, which rounds to 75.4
    3. Lateral area: 2 × π × 2 × 6 = 75.398223…, which rounds to 75.4
    4. Total surface area: 2 × 12.566370… + 75.398223… = 100.530964…, which rounds to 100.53

    The row where two outputs are equal: the volume and the lateral area are both 75.4. That is not a typo and it is not a special case — setting πr²h equal to 2πrh leaves r = 2 for every height, so at this radius the volume and the side area are the same number whatever height you choose. The units are not the same, cubic against square, and the agreement is purely numerical. It is the sort of thing a reader spots on the results panel and writes in about, so it is better met here first.

  5. A radius of 1 and a height of 1

    1. Base area: π × 1 × 1 = 3.141592, which rounds to 3.14
    2. Volume: 3.141592 × 1 = 3.141592, which rounds to 3.14
    3. Lateral area: 2 × π × 1 × 1 = 6.283185…, which rounds to 6.28
    4. Total surface area: 2 × 3.141592 + 6.283185… = 12.566370…, which rounds to 12.57

    The same coincidence at a different pair of columns: with a height of one the volume equals the base area, because multiplying by a height of one changes nothing. Again it holds at any radius and again only the numbers agree, not the units. Taken with the row above it makes a point worth holding on to — when two columns of a table match, the first thing to check is whether an input happens to sit at a value that forces it, and the second is whether the two columns are even measuring the same kind of thing.

Limitations

The outputs are always cubic and square centimetres, whatever unit the input dropdown is set to. Choosing inches changes how your two numbers are interpreted on the way in and nothing about how the answer is shown, so a cylinder entered as 2 inches by 3 inches comes back as 38.61 cubic centimetres and you have to convert it back yourself. That is deliberate and the reference table follows the same rule so the page never contradicts itself. The radius and height are read as a radius and a perpendicular height, and the page cannot tell if what you measured was the diameter or the distance across the curved side, so both of those go in as given and produce a plausible-looking wrong answer. Two decimal places is a display width and not a claim about precision: these are measurements of real objects, and the last digit is not meaningful for anything except an exact input. The total surface area is computed from the unrounded radius, so adding the printed base area to the printed side area can differ from the printed total by a hundredth — the printed total is the one to trust. The page assumes the cylinder is closed at both ends; a pipe or an open can has less surface area than the figure shown, by exactly one base area. And it takes no account of wall thickness, so a tank's capacity is the inside volume and the material needed is the outside surface, two measurements that the dropdown cannot express at once.

Frequently asked questions

Which two measurements do I need?
The radius of one of the circular ends and the perpendicular height between the ends. The radius is half the way across, so a cylinder measured across the top comes to 5 from a 10 centimetre tin. The height is measured straight up the side rather than across the curved surface, and the difference matters: on a radius of 5 and a height of 12 the straight distance is 12 but the shortest path across the outside is longer, and using the longer one overstates the volume.
Why is the answer in centimetres when I typed in inches?
Because the four outputs are a fixed set in centimetres, chosen so the results panel and the reference table below it always agree. The dropdown changes how your numbers are understood on the way in, not how the answer is written out. If you need the answer in your own unit, divide the volume by 16.387 to get cubic inches and the three areas by 6.4516 to get square inches.
What is the difference between the lateral area and the surface area?
The lateral area is the curved side on its own. The total surface area is the curved side plus both circular ends. For a closed cylinder the total is the one you want; for a pipe, an open drum or anything without ends, the lateral area is the one you want. The page prints both, plus the base area, so that a question about the ends can be answered without subtracting anything.
How do I turn the volume into litres?
Divide by a thousand. One cubic centimetre is one millilitre and a thousand millilitres make a litre, so a volume of 942.48 cubic centimetres is 0.942 litres. The conversion is exact in both directions and is the reason this page does not need a separate capacity mode — the millilitre was defined as a cubic centimetre in the first place.
Why do the volume and the lateral area come out the same?
Only when the radius is 2. Setting the volume πr²h equal to the lateral area 2πrh and cancelling leaves r = 2, whatever the height is, so at that radius the two columns match on every row. The same thing happens between the volume and the base area when the height is 1. The two figures are equal in value only: one is a volume and one is an area, and they are measured in different units.
Does the page work for a pipe or an open can?
For the volume and the lateral area, yes, and those are the two figures a pipe or an open can actually needs. The total surface area assumes both ends are closed, so for an open container subtract one base area from it. Nothing in the page knows about wall thickness either, so a tank's capacity is an inside measurement and its material is an outside one, and the two would need different radii.

References

Related calculators

Cone Volume CalculatorVolume, slant height and surface area of a cone.Surface Area CalculatorSurface area of spheres, cones, cylinders and prisms.Sphere Volume CalculatorHow much space a sphere takes, from its radius or diameter.Cuboid Volume CalculatorVolume and surface area of a rectangular box.