The volume and surface area formulas
V = base area × height
A = 2 × base area + perimeter × height
Where the formula comes from
A prism is a flat shape given thickness, so its volume is the area of that shape times how thick it is. This holds for every prism regardless of base — triangular, hexagonal, irregular. The lateral surface is the base perimeter times the height, because unrolling the sides gives one long rectangle.
A worked example, step by step
A hexagonal prism with sides of 4 cm and a height of 10 cm
- Apothem: 4 ÷ (2 tan(180°/6)) = 3.46 cm.
- Base area: 6 × 4 × 3.46 ÷ 2 = 41.57 cm².
- Volume: 41.57 × 10 = 415.69 cm³.
- Lateral surface: 24 × 10 = 240 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
Structural beams, hexagonal pencils and honeycomb packing, where the hexagon tiles a plane with the least material. Most manufactured extrusions are prisms, which is why the formula turns up constantly in materials estimating.