Prisms with regular bases
A right regular prism has two identical bases that are regular polygons, with all sides and angles equal, joined by rectangular side faces perpendicular to the bases. A triangular prism, a square prism or cube, a hexagonal pencil and the cells of a honeycomb are all examples.
The formulas
For a base with n sides of length s, and a prism of height h:
- Apothem (center to the middle of a side): a = s ÷ (2 × tan(180°/n))
- Base area: A = n × s² ÷ (4 × tan(180°/n)) = perimeter × apothem ÷ 2
- Volume: V = A × h
- Lateral area: n × s × h
- Total surface area: n × s × h + 2A
Example
A hexagonal prism with sides of 2 and a height of 5:
- Apothem: 1.732
- Base area: 6 × 4 ÷ (4 × tan 30°) = 10.392
- Volume: 10.392 × 5 = 51.96
- Lateral area: 6 × 2 × 5 = 60
- Total surface area: 60 + 2 × 10.392 = 80.78
Results are in the units you used for the lengths: square units for areas and cubic units for the volume.
Common cases
- Triangular prism (n = 3): base area = √3 ÷ 4 × s².
- Square prism (n = 4): base area = s²; with h = s, it is a cube.
- Hexagonal prism (n = 6): base area = 3√3 ÷ 2 × s², the shape of nuts, pencils and basalt columns.
- As n grows, the prism approaches a cylinder whose radius is the apothem.
The same approach applies to any prism: the volume is always the base area times the height, even when the prism is oblique, as long as h is the perpendicular height.