How this instrument works
A pyramid's volume is exactly one-third of the prism that shares its base and its height — V = ⅓Bh, where B is the base's area and h is the perpendicular distance from that base up to the apex. The ⅓ is not an approximation or a rule of thumb; it is a fixed ratio that holds no matter whether the base is a triangle, a square, a regular hexagon, or an irregular field boundary, because every cross-section of the pyramid, taken parallel to the base, shrinks by the same proportion as you rise toward the apex.
The proof is old and elegant. Eudoxus of Cnidus showed by the method of exhaustion, around 350 BCE, that three pyramids of equal base and height can be assembled from a single triangular prism, so each one occupies exactly a third of it. Two thousand years later, Cavalieri's principle restates the same idea in modern terms: match the cross-sectional areas of two solids at every height and their enclosed volumes match too — a shortcut that lets ⅓Bh cover every base shape at once, including a circle, where the pyramid becomes a cone.
One limit worth noting: as the base area shrinks toward zero while the height stays fixed, the volume falls straight to zero with it — there is no minimum pyramid. And doubling only the height, holding the base fixed, doubles the volume exactly, a linear relationship that a lot of people expect, wrongly, to also hold when a radius or side length changes, where volume instead scales with the square or the cube of that dimension.
- Enter the pyramid's base area into the Base area field — if you only have side lengths, multiply them first, or use the shape's own area formula for a triangular or hexagonal base.
- Enter the vertical distance from the base plane to the apex into the Height field — this is the perpendicular height, not the slant height running down a triangular face.
- Read the result in the Volume field: this instrument applies V = ⅓ × Base area × Height and returns the answer immediately.
- Keep both inputs in the same length unit; the volume that comes back will be in that unit, cubed.
Worked example — base area 36, height 5
Take a pyramid whose base measures exactly 36 square units in area, standing 5 units tall from that base to its apex. V = ⅓ × 36 × 5 = ⅓ × 180 = 60 cubic units — a clean number because 36 is itself a multiple of 3, so the one-third factor divides evenly with nothing left over.
Check it against the prism it sits inside: a prism with the same 36-unit base and the same 5-unit height holds 36 × 5 = 180 cubic units, and the pyramid occupies exactly a third of that enclosure — 60 of the 180. That one-third relationship is fixed regardless of whether the base is a square, a triangle, or a circle standing in for a cone.
Questions
What is the formula for the volume of a pyramid?
V = ⅓ × B × h, where B is the area of the base and h is the perpendicular height to the apex. The formula holds for any base shape — triangle, square, hexagon, or an irregular polygon — because the proof compares cross-sectional areas at each height rather than assuming a particular outline.
Why does the formula include a factor of one-third?
Because three pyramids of equal base and height pack exactly into one prism of that same base and height, a fact Eudoxus proved by the method of exhaustion around 350 BCE. Cavalieri's principle later generalized it: match the cross-sectional areas of two solids at every height and their volumes match too, which is why ⅓Bh covers every base shape without adjustment.
Does this formula also apply to a cone?
Yes. A cone is a pyramid whose base has been rounded into a circle, so V = ⅓Bh still applies with B = πr². The same one-third relationship to the enclosing cylinder holds for exactly the same reason it holds between a pyramid and its enclosing prism.
What is the most common mistake people make with this formula?
Measuring the slant height instead of the perpendicular height. The slant height runs down a triangular face from the apex to the midpoint of a base edge, while h in V = ⅓Bh has to be measured straight down from the apex to the base plane — for anything steep, the two figures differ enough to throw the volume off.
How does this relate to the height-of-cone and height-of-cylinder calculators on this site?
Those pages start from a known volume and solve backwards for the height. This page runs the same relationship, V = ⅓Bh, in the forward direction — from a measured base area and height to the volume — which is why the labels differ even though the underlying equation is identical.