How this instrument works
A square pyramid's volume follows the standard pyramid rule shared by every pyramid regardless of base shape: one third of the base area multiplied by the height. Since a square base's own area is simply s², the formula becomes V = s²h ⁄ 3. This page returns just that single figure, for whenever only the volume is wanted, a narrower and quicker route than a combined solver that also computes the full surface area alongside it.
The one-third factor isn't arbitrary: three pyramids sharing an identical base and height can be assembled, with no gaps or overlaps, to exactly fill a rectangular box (a prism) of that same base and height — a genuine geometric fact, provable through coordinate integration, that holds for every pyramid shape, square-based or otherwise.
Doubling the base edge s while holding the height fixed doesn't merely double the volume — it quadruples it, since s appears squared in the formula, while doubling the height alone does simply double the volume, since h appears only to the first power. The two dimensions affect the total very differently.
- Enter the square base's edge length into the Base edge length field.
- Enter the pyramid's height (apex to base plane, measured perpendicularly) into the Height field.
- Read Volume: the sheet applies s²h ⁄ 3 directly.
Worked example — base edge 4, height 6
A square pyramid has a base edge of 4 and a height of 6. Its volume is 4² × 6 ⁄ 3 = 16 × 6 ⁄ 3 = 32 cubic units — the base's own area, 16, carried through the standard one-third pyramid rule.
A larger pyramid with base edge 5 and height 9 has a volume of 25 × 9 ⁄ 3 = 75 cubic units — reached through both a wider base and a taller height than the golden example, illustrating how the two dimensions compound together in the final total.
Questions
What is the formula for the volume of a square pyramid?
V = s²h ⁄ 3, where s is the square base's edge length and h is the perpendicular height. The s² term is simply the base's own area, carried through the standard one-third pyramid-volume rule.
Why is there a division by 3?
Because exactly three pyramids sharing an identical base and height can be assembled, with no gaps or overlaps, to fill a rectangular box of that same base and height — a general geometric fact true for any pyramid, provable through coordinate integration.
How is this different from a combined square pyramid solver?
A combined solver returns both surface area and volume together. This page focuses on just the volume, a quicker route whenever surface area isn't part of what's needed for the task at hand.
Does doubling the base edge double the volume?
No — the base edge appears SQUARED in the formula, so doubling it quadruples the volume. Doubling the height alone, by contrast, does simply double the volume, since height appears only to the first power.
What if the base edge is zero?
The volume is exactly zero regardless of the height — a base edge of zero collapses the pyramid to a single point with no base area at all.