V = πr²h — base area times height, with nothing to divide by. A cylinder of radius 3 and height 5 holds 45π. Its lateral area is 30π and its total area is 48π, and which one you want depends on whether the ends are there.
Which area you want
“The surface area of a cylinder” is ambiguous, and the ambiguity costs real money.
Lateral area is the curved wall alone: 2πrh. This is what you want for a pipe, a column, a label wrapping round a tin, or the paint on a silo.
Total area adds both circular ends: 2πrh + 2πr². This is what you want for a sealed can or a closed tank.
The gap is 2πr², which is small for a long thin cylinder and dominant for a short wide one. For a disc-shaped cylinder the ends can be most of the surface, so picking the wrong one is not a rounding matter.
This page reports both rather than choosing for you.
Unrolling the wall
The lateral area needs no memorising, because the wall unrolls into a plain rectangle.
Its width is the circumference, 2πr. Its height is the cylinder’s, h. So the area is 2πrh, and that is the whole derivation.
It is why a tin’s label is a rectangle, and why the label’s length must match the circumference rather than the diameter — a distinction that catches people out by a factor of π.
A cylinder is a developable surface: it can be made from flat material without stretching. So is a cone. A sphere is not, which is why a globe cannot be flattened into a map without distortion.
A third for a cone
A cone with the same base and the same height holds exactly a third as much. This page shows that figure alongside so the relationship is visible.
The factor is a third for every cone and every pyramid, whatever the base shape — which is the clue that it has nothing to do with circles. It comes from Cavalieri’s principle and the squared scaling of a cross-section.
A hemisphere of the same radius sits between them, at two thirds of a cylinder of height r. Stacking the three — cone, hemisphere, cylinder — in the ratio 1 : 2 : 3 is one of the neater facts in solid geometry, and it is Archimedes’ result restated.
The least-material shape
For a fixed volume, which cylinder uses the least material? The answer is clean, and what happens in practice is more interesting.
The optimum has height exactly equal to diameter — the cylinder that would just contain a sphere. This page reports those dimensions for whatever volume you enter.
Almost no real can is that shape. A drinks can is much taller and narrower than the optimum, and the reason is that the calculation above assumes uniform thickness. The top and bottom of a can are thicker gauge than the wall, because they take the pressure and the seaming, so the true optimum shifts towards a taller shape.
Handling matters too. A can has to be comfortable to hold and to fit a vending mechanism, and those constraints are worth more than a few percent of aluminium.
It is a good example of a textbook optimisation whose answer is right and whose conclusion is wrong, because the model left something out.
Where it gets used
Tanks and vessels. Most storage is cylindrical: it is easy to fabricate from rolled sheet and handles pressure far better than a box.
Pipes. Flow depends on the cross-sectional area, which goes as r² — so doubling a pipe’s diameter quadruples its area, and rather more than quadruples its flow.
Engines. Displacement is the swept cylinder volume times the number of cylinders, which is what an engine’s stated capacity means.
Packaging. Cans, bottles and drums, where the trade between material cost and handling shapes every one of them.
Sources and methodology
The cylinder formulas are standard geometry; these are the references.
Method. Lateral and total surface area are reported as separate figures rather than one being presented as the surface area, because the two answer different practical questions and differ by 2πr² — a large gap for a short wide cylinder. The cone of the same base and height is shown alongside so the exact one-third relationship is visible, and the suite asserts that third on four hundred generated pairs. That engine is verified on every change against 96 hand-written assertions, including that a cone is exactly a third of its cylinder across four hundred generated radius and height pairs. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The cone that holds exactly a third of this cylinder is on the Cone Calculator.