A pyramid with its top cut off
A frustum of a pyramid is what remains when a pyramid is cut by a plane parallel to its base and the top is removed. A regular frustum has two similar regular polygons as its bases, the larger one at the bottom and the smaller one at the top, and its lateral faces are identical isosceles trapezoids. Hoppers, planters, lampshades and concrete footings often take this shape.
The formulas
For a regular frustum with n sides, bottom side a1, top side a2 and height h:
- Base areas: A = n × a² ÷ (4 × tan(180° ÷ n)) for each base
- Volume: V = h ÷ 3 × (A1 + A2 + √(A1 × A2))
- Apothem of each base: r = a ÷ (2 × tan(180° ÷ n))
- Slant height: s = √(h² + (r1 − r2)²)
- Lateral area: n × (a1 + a2) ÷ 2 × s
- Lateral edge: √(s² + ((a1 − a2) ÷ 2)²)
- Total area: lateral area + A1 + A2
Example
A square frustum with a 6 cm bottom, a 3 cm top and a height of 4 cm:
- Base areas: 36 cm² and 9 cm²
- Volume: 4 ÷ 3 × (36 + 9 + √324) = 4 ÷ 3 × 63 = 84 cm³
- Slant height: √(4² + 1.5²) = 4.272 cm
- Lateral area: 4 × (6 + 3) ÷ 2 × 4.272 = 76.896 cm²
- Total area: 121.896 cm²
- Lateral edge: 4.528 cm
A formula from ancient Egypt
The Moscow Mathematical Papyrus, written about 1850 BC, contains a problem asking for the volume of a square frustum with a bottom of 4, a top of 2 and a height of 6. The scribe’s method matches the modern formula and gives 56, one of the oldest known results in solid geometry.
Checking the result
The frustum volume always lies between the volume of a prism with the top base, A2 × h, and a prism with the bottom base, A1 × h. When a2 equals a1 the formula gives a prism, and when a2 is zero it gives a full pyramid, V = A1 × h ÷ 3.