Cube Surface Area

Six identical square faces, so $S = 6a^2$. Painting a cubic tank 2 m on a side means covering $6 \times 2^2 = 24$ m², which at 10 m² per litre is about 2.4 litres per coat. Backwards, $a = \sqrt{S/6}$ takes the principal positive root, an edge length being positive by definition: 54 m² of sheet metal folds into a cube 3 m on a side.

Set this beside $V = a^3$ and something important falls out. The ratio of surface to volume is $S/V = 6/a$, which shrinks as the cube grows. Skin goes up with the square of size while contents go up with the cube, so a big object has proportionally less outside for its inside. That single fact explains a long list of otherwise unrelated observations: why a large block of ice outlasts the same mass in cubes, why bulk storage costs less per litre to insulate, why a mouse must eat constantly while an elephant does not, and why cells stay microscopic. Past a certain size, a cell cannot get nutrients across its membrane fast enough to supply its own interior.

The cube also holds a modest optimisation prize. Of all rectangular boxes with a given volume, the cube has the least surface area, so it is the cheapest box to build, to clad and to heat. A sphere beats it outright, that is why bulk gas is stored in spheres, but among shapes you can make from flat sheet and square corners, the cube wins.

Three things go wrong. The first is counting faces you do not have. A tank sitting on a slab presents five faces to the paintbrush, not six; an open-topped bin has five as well but a different five; a form for casting has an inside and an outside. Ask which surfaces actually exist before multiplying by 6. The second is the unit exponent: areas convert by the square of the length factor, so 1 m² is 10,000 cm², not 100, and an edge given in centimetres with an answer wanted in square metres needs converting first. The third is expecting the surface area to scale with the volume: it does not, and the whole previous paragraph is about why. Doubling the edge quadruples the paint and octuples the contents, so a coating estimate borrowed from a smaller tank of the same shape will always run high per litre stored.