Cylinder Surface Area

Take a can, cut the two ends off, slit the side and lay it flat. What you have is two discs and one rectangle — and the rectangle's width is the distance the cut travelled around the can, which is the circumference $2\pi r$. That is the whole derivation: $S = 2\pi r^2 + 2\pi r h$, two caps plus a label. The second term is called the lateral or curved surface area, and it is the one that does most of the work in practice, because it is the part that grows with the height.

A worked instance in units you would meet. A vertical hot water storage tank 600 mm in diameter and 1.5 m tall has $r = 0.3$ m, so the caps come to $2\pi(0.3)^2 = 0.57$ m² and the side to $2\pi(0.3)(1.5) = 2.83$ m², totalling 3.39 m². That is the jacket area, and multiplied by the temperature difference and divided by the insulation's R-value it is the tank's standing heat loss. Backwards, $h = S/(2\pi r) - r$ tells you how tall a tank of a given radius must be to reach a required surface — the subtraction of $r$ is the caps being paid for first.

Read alongside the volume $V = \pi r^2 h$, this formula answers a design question. For a fixed volume, the surface area is smallest when $h = 2r$ — when the can is exactly as tall as it is wide. That is the shape that uses the least metal per litre, and it is not the shape of the can in your cupboard, because the ends need thicker stock and a seaming operation the wall does not, so real cans are made taller and narrower than the geometry alone would suggest. Cost per square centimetre is not uniform, and the optimum shifts accordingly.

Three ways the answer comes out wrong. The first is the perennial one: diameter entered as radius, which quadruples the caps and doubles the side. The second is not asking which surfaces actually exist. This formula is for a closed cylinder. An open-topped drum has one cap, not two, so subtract $\pi r^2$; a bare length of pipe has no caps at all and is $2\pi r h$ alone; and a pipe you intend to coat inside and out has two curved surfaces at slightly different radii. The third shows up when solving for height: if the surface area you enter is less than the two end caps could account for on their own, no positive height exists and the solver says so rather than returning a negative tank.