Frustum of a pyramid: the cut-off top

A frustum is what remains when a smaller similar pyramid is sliced off the top.

Key formulas

You see two parallel bases with areas A₁ and A₂ and a perpendicular height h between them. The solid is neither a full pyramid nor a prism.

A₁ = 36, A₂ = 16, h = 9. Then √(A₁A₂)=24. V = (1/3)×9×(36+16+24)= (3)(76)=228.

The √(A₁A₂) term comes from comparing the large pyramid with the small one that was removed. Skipping it underestimates the volume.

Problem. A frustum has parallel face areas 36 cm² and 16 cm² and height 9 cm. Find its volume.

Solution. √(A₁A₂) = √(36 × 16) = 24. V = (1/3)h(A₁ + A₂ + √(A₁A₂)) = (1/3)×9×(36+16+24) = 3 × 76 = 228 cm³.

If A₁=A₂, the formula becomes A×h, the prism case — a good check.

Frustum areas 25 and 9, height 6. Volume is:

They must be similar shapes lying in parallel planes.