The volume and surface area formulas
V = a³
A = 6a²
Where the formula comes from
The volume of any prism is the base area times the height. In a cube all three dimensions are the same edge, so a × a × a gives a³. The surface is six faces, each a square of side a, hence 6a². The interior diagonal comes from applying Pythagoras twice: once across a face and once from that face diagonal up to the opposite corner.
A worked example, step by step
A cube with an edge of 3 cm
- Volume: 3³ = 27 cm³.
- Surface: 6 × 3² = 6 × 9 = 54 cm².
- Sum of the edges: 12 × 3 = 36 cm.
- Interior diagonal: 3 × √3 = 5.20 cm.
Watch the units: volume goes in cubic units and surface area in square ones. Mixing them is the most common mistake.
Where it turns up in practice
Packaging and shipping, where cubic boxes are the cheapest to make and stack; dice and puzzle design; and estimating material for anything cast or moulded. The cube is also the reference shape for the litre, since a cube 10 cm on a side holds exactly one litre.