Last updated: 2026-06-06

The Pyramid Frustum Defined

A pyramid frustum — also called a truncated pyramid — is the solid that remains when the top of a pyramid is removed by a cut parallel to the base. It has two parallel polygonal faces (a larger bottom base with area A₁ and a smaller top base with area A₂), connected by trapezoidal lateral faces, and a vertical height h measured perpendicularly between the two bases.

Unlike the cone frustum, where the bases are always circles, a pyramid frustum accepts any polygonal base: square, rectangular, triangular, hexagonal, or any other shape, provided both bases are similar (same proportions, different scale).

Formula

The volume of a pyramid frustum is given by the prismatoidal formula:

$V = \frac{h}{3}\!\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)$

where A₁ is the bottom base area, A₂ is the top base area, and h is the perpendicular height between the bases.

The three terms — A₁, A₂, and √(A₁A₂) — represent the bottom area, top area, and their geometric mean, respectively. Each is weighted equally by h/3.

Why the Three-Term Formula

The formula follows from the cone-of-similar-solids construction. Consider a full pyramid with apex height H above the bottom base and base area A₁. The small pyramid cut from the top has height H − h and base area A₂. By the similar-solid scaling law, areas scale as the square of the linear scale factor, so:

$\frac{A_2}{A_1} = \left(\frac{H - h}{H}\right)^2$

Solving for H and substituting into the volume difference:

$V = \frac{A_1 H}{3} - \frac{A_2 (H - h)}{3}$

After eliminating H algebraically and simplifying, the expression reduces exactly to the three-term form above. The geometric mean √(A₁A₂) appears because it is the area of the intermediate cross-section at half-height when the pyramid is extended.

Limiting Cases

Two degenerate cases serve as sanity checks for any frustum calculation:

Complete pyramid (A₂ = 0): The formula becomes V = A₁·h/3, the standard pyramid volume formula. The geometric mean term vanishes.

Prism (A₁ = A₂ = A): The formula becomes V = (h/3)(A + A + A) = A·h, the standard prism volume formula. This confirms that a frustum with equal top and bottom areas is simply a prism.

Worked Example

A concrete planter has a square bottom face 30 cm × 30 cm and a square top face 20 cm × 20 cm, with a height of 25 cm.

• A₁ = 30 × 30 = 900 cm²
• A₂ = 20 × 20 = 400 cm²
• √(A₁·A₂) = √(900 × 400) = √360,000 = 600 cm²
• V = (25/3)(900 + 400 + 600) = (25/3) × 1,900 ≈ 15,833 cm³ ≈ 15.8 litres

The planter holds roughly 15.8 litres of soil — about the same as a standard 15-litre potting bag.

Using Area Inputs Directly

This calculator accepts base areas directly rather than individual side lengths. This approach makes the formula general across all base shapes: enter the area of a hexagonal frustum, a triangular frustum, or any other polygon without needing a separate formula for each shape. For a square frustum with side length s, enter A = s². For a regular hexagon with side a, enter A = (3√3/2)a².

Related Calculators

• Pyramid Volume Calculator — volume, surface areas, and slant height of a complete pyramid
• Cone Frustum Calculator (Truncated Cone) — volume and surface areas of a truncated cone
• Rectangular Prism Calculator — volume and surface area of a box (the A₁ = A₂ special case)