Pyramids with regular bases
A regular pyramid has a base that is a regular polygon, with equal sides and angles, and an apex directly above the center of the base. Its side faces are identical isosceles triangles. The Great Pyramid of Giza, a square pyramid, is the most famous example, but tents, roofs, spires and gemstones use the same shape.
The formulas
For a base with n sides of length s, and a pyramid of height h:
- Apothem of the base: a = s ÷ (2 × tan(180°/n))
- Base area: A = n × s² ÷ (4 × tan(180°/n))
- Volume: V = A × h ÷ 3
- Slant height, from the apex to the middle of a base side: l = √(h² + a²)
- Lateral area: n × s × l ÷ 2
- Total surface area: lateral area + A
- Lateral edge, from the apex to a base corner: √(h² + R²), where R = s ÷ (2 × sin(180°/n)) is the circumradius of the base
Example
A square pyramid with a base side of 6 and a height of 4:
- Base area: 36, apothem 3
- Volume: 36 × 4 ÷ 3 = 48
- Slant height: √(4² + 3²) = 5
- Lateral area: 4 × 6 × 5 ÷ 2 = 60
- Total surface area: 96
- Lateral edge: 5.83, and each face rises at 53.1° from the base
Results are in the units used for the lengths: square units for areas and cubic units for the volume.
The one-third rule
A pyramid holds exactly one third of the prism with the same base and height. Three identical oblique square pyramids, each with its apex above a corner of its base, even fit together into a cube. The rule holds for any pyramid or cone, which is why a cone’s volume is πr²h ÷ 3.
Giza for scale
The Great Pyramid originally had a base side of about 230 m and a height of about 147 m. Its faces rise at about 52°, and its volume was roughly 2.6 million cubic meters.