Pyramid Frustum Volume

Volume of a frustum of a pyramid

Slice the top off a pyramid with a plane that runs parallel to the base, and the leftover chunk is a pyramidal frustum. The same volume formula works no matter what shape the base happens to be (square, triangular, hexagonal): V = (h/3)·(A₁ + A₂ + √(A₁·A₂)). Here A₁ is the area of the larger base, A₂ is the smaller top, and h is the perpendicular height separating the two planes.

Take a base of 100 m², a top of 25 m², and a height of 6 m. That gives V = (6/3)·(100 + 25 + √(100·25)) = 2·(100 + 25 + 50) = 350 m³. Push A₂ toward 0 and the formula reduces to V = A₁·h/3, the volume of a full pyramid. Set A₁ equal to A₂ and you get V = A·h, a prism.

Applications

You run into this shape more often than you'd think. The Great Pyramid of Khufu was a frustum at every construction stage before it was capped. So are pyramid-shaped lampshades (abat-jour), filled-in or excavated terrain, sand and grain stockpiles that taper toward the top, and structural footings that fan out wider as they reach the ground.

FAQ

Does the base shape matter? It doesn't. The formula cares only about the two parallel area values and the height, so square, triangular, and irregular bases all run through the same expression.

What if I only have side lengths? Work out A₁ and A₂ with the polygon area formula first, and feed those into the frustum formula.

Is h the slant height? No. It's the perpendicular distance between the two parallel planes, not the length of the slanted edge.

Why the square root term? That term is the geometric mean of the two areas, and it accounts for how the cross-section tapers linearly from one face to the other. Drop it, and your answer comes out wrong for any frustum where A₁ ≠ A₂.