Cuboids and cubes

Two different questions about a box:

• surface area: how much paper wraps the outside? (open out a sweet box flat and you see all six faces);
• volume: how much fits inside?

A cuboid has six rectangular faces, in three equal pairs, like the floor and ceiling, front and back wall, left and right wall of a room. The floor and ceiling take the same amount of paint. So total surface area = 2(lw + wh + lh).

Volume = the number of unit cubes inside = l × w × h. Or think of a stack of thin cards: volume = area of the base × height. 216 cm³ means 216 cubes of 1 cm each would fill it.

A cube of side a: surface area 6a², volume a³. So a cube of volume 64 cm³ has side 4 cm and surface area 96 cm².

Same volume, different area. A 6 cm cube and a 9 × 6 × 4 cuboid both hold 216 cm³, but the cuboid needs 228 cm² of wrapping against the cube's 216 cm².

Cutting adds surface. A 4 cm cube cut into 1 cm cubes keeps its volume, but its total surface area grows 4 times: every cut makes two new faces.

A neat trick. If the three faces at a corner have areas lw, wh and lh, their product is (lwh)². So areas 6, 15 and 10 give a volume of √900 = 30.