Cylinder Lateral Surface Area

Cut a cylinder's wall along a vertical line and roll it flat and you get a rectangle: $2\pi r$ wide, $h$ tall, so $A = 2\pi r h$. No ends, no caps, just the wrap. This is the paper label on a soup can, the sheet metal around a round duct, the pipe insulation you order by the square metre, and the area a painter charges for on a storage silo.

Keep it distinct from the closed cylinder's total surface $2\pi r^2 + 2\pi r h$, which adds the two circular ends. For a squat shape the difference is enormous. A cylinder with $r = 1$ m and $h = 0.2$ m has a wall of only 1.26 m² but end caps totalling 6.28 m², so quoting the total when you meant the wrap overstates the job by a factor of six. For a long thin pipe the ends are negligible and the two numbers converge.

The inverse is the one field crews actually reach for: given a roll of jacket of known area, how far along the pipe will it get you? That is $h = A/(2\pi r)$, and it is why insulation is sold by area rather than by length.