Square Pyramid Volume Calculator
A square pyramid is a solid with a square base and four triangular faces meeting at a single apex above the centre of the base. Its volume is exactly one-third of the volume of a box that shares the same square base and height. This tool computes that volume from the base edge length and the perpendicular height using the standard solid-geometry formula. It also reports the base area and the equivalent prism volume so you can see the one-third relationship directly. Keep both length inputs in the same unit to get a correct cubic-unit result.
Square pyramid volume formula
Base area = a * a (a squared)
Volume = (1 / 3) * base area * h
Equivalent prism volume = base area * h
Volume = prism volume / 3
Here a is the length of each square base edge and h is the perpendicular height from the base to the apex. The volume is always exactly one-third of the prism that shares the same base and height.
How square pyramid volume works
- The one-third factor is exact for every pyramid, a result that follows from integral calculus and from dissecting a cube into three congruent pyramids.
- The height must be the perpendicular (vertical) height, not the slant height along a triangular face.
- The base area for a square base is the edge length squared.
- Doubling the height doubles the volume; doubling the base edge quadruples the volume because area scales with the square of length.
- The volume is reported in cubic units matching the unit of your inputs, so keep both inputs in the same unit.
Square pyramid volume: frequently asked questions
What is the formula for the volume of a square pyramid?
The volume of a square pyramid equals one-third of the base area times the height: V = (1/3) * a^2 * h, where a is the length of each side of the square base and h is the perpendicular height from the base to the apex. This is a special case of the general pyramid volume formula V = (1/3) * base area * height.
Does the height mean the slant height or the vertical height?
This calculator uses the perpendicular (vertical) height, measured straight up from the centre of the base to the apex. It is not the slant height, which runs along the triangular face. If you only know the slant height, you must first convert it to vertical height using the Pythagorean theorem before using this tool.
What units does the result use?
The volume is expressed in cubic units of whatever unit you enter. If you enter the base edge and height in centimetres, the volume is in cubic centimetres. If you enter metres, the volume is in cubic metres. Keep both inputs in the same unit for a correct result.
Why is the volume one-third of a prism with the same base?
A pyramid occupies exactly one-third of the volume of a prism (or box) sharing the same base and height. This one-third ratio is a fundamental result of solid geometry, provable by integration or by dissecting a cube into three congruent pyramids, and holds for any pyramid regardless of base shape.
Can I use this for a rectangular pyramid?
No. This calculator assumes a square base where all four edges are equal. For a rectangular base with different length and width, multiply length times width for the base area instead of squaring a single edge, then take one-third of that area times the height.
Official sources
- National Institute of Standards and Technology: SI Units and measurement.
- NASA Glenn Research Center: Volume of solids reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.