V = base area × vertical height ÷ 3. A square pyramid with base 6 and height 4 holds 48. Its slant height is 5 and its lateral edge is √34 — three different distances from the apex, and the formulas want different ones.
Three distances
There are three ways to measure from the apex to the base, and they are all different numbers.
Vertical height. Straight down to the centre of the base. This is what the volume formula wants, and it is the shortest of the three.
Slant height. Down the middle of a triangular face, to the midpoint of a base edge. This is what the lateral area wants.
Lateral edge. Along the corner, from apex to a base vertex. This is the longest, and it is what you would measure with a tape laid on the outside.
For a square pyramid with base 6 and height 4: vertical 4, slant 5, lateral edge √34 ≈ 5.83. Three quite different numbers, and substituting one for another gives a plausible wrong answer with no warning at all.
Two slant heights
This is the part most pages get wrong.
A square base has one slant height, because all four faces are identical.
A rectangular base has two. The faces over the long edges lean at a different angle from the faces over the short edges, so they have different slant heights, and the lateral area is the sum of two different products.
A calculator offering a single “slant height” box therefore cannot be right for a rectangular base. It will produce an answer, and the answer will be wrong.
This page derives both from the base dimensions and the vertical height, and shows both whenever they differ.
Why a third
A pyramid holds exactly a third of the prism with the same base and height. Exactly, and for every base shape.
The same third applies to a cone and its cylinder, which is the clue that the factor has nothing to do with squares or circles specifically.
It comes from Cavalieri’s principle: two solids whose cross-sections match at every height have the same volume. A pyramid’s cross-section at height x is the base scaled by (1 − x/h), so its area carries a squared factor, and integrating that square across the height produces the third.
There is a physical demonstration for the cube: three identical square pyramids, each with the cube’s base and half its height, fit together to fill it exactly. It is a satisfying thing to make from card.
The regular tetrahedron
Four equilateral triangles, four vertices, six edges. The simplest polyhedron there is, and the only one where every face touches every other.
Its volume is a³/(6√2) and its height is a√(2/3) — and that height is not the height of one of its triangular faces, which is a distinction worth keeping when working from a net.
Its dihedral angle — the angle between two faces — is arccos(1/3) ≈ 70.53°. That number has a consequence: five tetrahedra around an edge come to about 352.5°, leaving a gap of 7.5°. Tetrahedra do not tile space.
Aristotle believed they did, and stated it. The error stood for about eighteen centuries before being corrected in the fifteenth. It is a good reminder that a plausible geometric claim can survive a very long time without anyone checking the arithmetic.
Where it gets used
Architecture. Pyramidal roofs, hip roofs and spires. The lateral area is what the covering material is based on, so the slant heights are the measurements that matter.
Storage and handling. Hoppers with rectangular openings are pyramidal frustums, and the volume calculation follows the same reasoning as a cone frustum.
Chemistry. Tetrahedral molecular geometry — methane, and every sp³-hybridised carbon — sits at the arccos(−1/3) ≈ 109.47° bond angle, which is the angle from the centre rather than the dihedral one.
Graphics and simulation. Tetrahedral meshes are the three-dimensional analogue of triangular ones, and finite element analysis is built on them.
Sources and methodology
The pyramid formulas are classical results; these are the references.
Method. The page asks for the vertical height and derives the slant heights and the lateral edge, so the three cannot be confused on the way in. A rectangular base yields two distinct slant heights and both are reported, because the lateral area needs both and a page offering one box is wrong for every non-square base. The suite asserts that a pyramid is exactly a third of the corresponding prism on four hundred generated cases. That engine is verified on every change against 96 hand-written assertions, including that a pyramid is exactly a third of its prism across four hundred generated base and height combinations, and that a rectangular base produces two genuinely different slant heights. The count and the per-case breakdown are published on the formula verification page.
Read the guide
A pyramid is exactly a third of the prism with the same base and height — the Prism Calculator handles that one.