Cone Lateral Surface Area

Cut a paper cone up one side and roll it flat and you get a sector of a circle: a pie slice whose radius is the slant height $l$ and whose curved edge is the base circumference $2\pi r$. A sector's area is half its arc times its radius, so $A = \ frac{1}{2}(2\pi r)(l) = \pi r l$. That is the whole derivation, and it is why sheet-metal workers can lay out a cone as a flat pattern before they ever bend anything.

The formula runs the trade backwards just as often. A fabricator with a fixed sheet size divides the available area by $\pi r$ to find the longest cone that can be cut from it, and a supplier quoting a run of conical hoppers prices the job by square metres of plate. Note what is missing: no base. This is the skin of the side only, which is exactly what you want for a witch's hat roof, a loudspeaker cone or a filter element.

The classic mistake is feeding in the vertical height instead of the slant height. They are different numbers, and the slant is always the larger of the two. A cone 3 m in radius and 4 m tall has a slant height of 5 m, so using 4 would undercount the metal by a fifth.