Rectangular Prism Surface Area

A box has six faces in three matching pairs, so the total area is $2(lw + lh + wh)$ — top and bottom, front and back, two ends. It is the formula behind every cardboard-cost estimate, every quote for cladding a duct, and every calculation of how much paint a shipping crate will drink. A carton 40 cm × 30 cm × 20 cm needs $2(1200 + 800 + 600) = 5200$ cm² of board, before you add a single glue flap.

The sanity check to keep in your pocket: set $l = w = h = s$ and the expression collapses to $6s^2$, the cube. If a general box formula does not reduce to six squares when you feed it a cube, it is wrong.

Solving backwards for one edge stays easy, which surprises people who expect a quadratic. Peel off the two faces that do not involve the unknown edge and what remains is linear: $A - 2wh = 2l(w + h)$. The catch is physical rather than algebraic. If the area you were given is less than those two fixed faces already account for, no box exists, and the solver says so rather than handing back a negative length.