Peel the label off a soup can and flatten it: a plain rectangle, as tall as the can and as long as the trip around the circle. That's the cylinder's lateral area — the wall alone: $A = 2\pi r h$ (read aloud: A equals two pi r h), where $r$ is the can's radius, $h$ its height, and $A$ the wall's area. Add the two lids, $\pi r^2$ each, for the total $S$: $S = 2\pi r^2 + 2\pi r h$. Read the job like a laser: a label wants the wall; a full paint job wants the lids too.

Cones bring one new character: the slant height $l$, the walk up the sloped side — always longer than the vertical height. Pythagoras delivers it: $l = \sqrt{r^2 + h^2}$ — l equals the square root of, r squared plus h squared — and then the cone's curved skin is $A = \pi r l$ — pi r l. Height is for standing; slant is for wrapping.