Pyramid Base Height Volume Calculator

Pyramid volume from base area and height

The volume of any pyramid is V = (1/3)·A_base·h, where A_base is the area of the base (in any shape — square, rectangle, triangle, hexagon, irregular polygon) and h is the perpendicular height from base to apex. Example: the Great Pyramid of Giza (Khufu) has a square base of 230 m and h ≈ 147 m, giving A_base = 52,900 m² and V = 52,900·147/3 ≈ 2.59 million m³. The factor 1/3 was proven rigorously by Cavalieri in the 17th century, but Democritus had stated it back in 440 BC: any pyramid fills exactly one third of the prism with the same base and height.

Applications

Archaeology (estimating stone mass and construction effort of pyramids), architecture (pyramidal roofs, atriums, the Louvre pyramid), packaging (TetraPak cartons are tetrahedra with triangular pyramid shape), 3D modelling and CAD, geology (volume of pyramidal mineral deposits and quarry stockpiles).

FAQ

Do I need to know the base shape? No — only its area. Compute A_base separately (square a², rectangle b·c, triangle b·h/2, hexagon 3√3·a²/2) and the volume formula stays the same.

Does it work for oblique pyramids? Yes. As long as h is the perpendicular height (apex projected onto the base plane), an oblique pyramid has the same volume as a right pyramid with the same base and height.

Why the 1/3 factor? Cavalieri's principle: cross-sections of the pyramid scale as (z/h)², integrating to h/3 — exactly one third of the corresponding prism.