Square pyramid ($V=a^{2}h/3$)

A square pyramid has a square base. If the base edge has length $a$ and all four lateral edges have length $\ell >a/\sqrt{2}$, the apex lies above the base center. The center-to-vertex distance is $a/\sqrt{2}$, so the Pythagorean theorem gives $h=\sqrt{{\ell }^2-a^2/2}$. Its volume is $a^{2}h/3$. When all eight edges have the same length $a$, this reduces to $a^3/(3\sqrt{2})$.