The volume and surface area formulas
V = 1/3 a²h
A = a² + 2a·l
Where the formula comes from
Three identical pyramids fit exactly inside a cube — a demonstration you can do with paper — which is where the one-third comes from. The slant height is the height of one triangular face, and it is √(h² + (a/2)²): a right triangle formed by the vertical height and half the base edge. Each face has area a·l/2, and there are four.
A worked example, step by step
A square pyramid with a base edge of 6 cm and a height of 4 cm
- Base area: 6² = 36 cm².
- Volume: 1/3 × 36 × 4 = 48 cm³.
- Slant height: √(4² + 3²) = 5 cm.
- Lateral surface: 4 × (6 × 5 ÷ 2) = 60 cm².
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
Roofs, marquees and hoppers, plus anything that needs to shed water from a square plan. Historically it is the shape of the Egyptian pyramids, whose builders clearly understood the one-third rule long before anyone wrote it down.