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.