the net of a solid
Take a cardboard box and carefully cut along enough edges to flatten it out without tearing any face. The flat pattern you get — a connected arrangement of the box's faces, ready to fold back up — is a net. Every cracker box, every juice carton, and every paper model started life as a net printed flat and then folded.
Formally, a net is a flat (two-dimensional) figure made of the faces of a polyhedron, joined along edges, that folds up to form the solid. A cube has 11 distinct nets; a square pyramid unfolds into a square with four triangular flaps; a cylinder unrolls into a rectangle (its curved side) plus two circles (its ends). Nets make surface area concrete: the surface area of a solid is just the total area of its net, because folding does not change how much material there is.
Not every arrangement of the right faces folds into the solid — fold lines and adjacencies must match — so a net is a specific valid pattern, not just any scattering of faces. Still, the net is the single most useful tool for seeing why a surface-area formula looks the way it does: flatten first, add up the flat pieces, and the formula falls out.
A closed cylinder of radius r and height h unrolls into two circles (each area pi·r^2) and a rectangle of width 2·pi·r (the circumference) and height h, giving total surface area 2·pi·r^2 + 2·pi·r·h.
Unfold, measure the flat pieces, add — that is surface area.
Not every layout of the correct faces is a valid net; the edges that meet when folded must match, so a cube has 11 nets, not arbitrarily many.