A regular octahedron can be viewed as two square pyramids base-to-base. The distance between opposite vertices (the apexes) equals $a\sqrt{2}$. This can be derived from the fact that the octahedron's vertices can be placed at $(\pm a/\sqrt{2}, 0, 0)$, $(0, \pm a/\sqrt{2}, 0)$, $(0, 0, \pm a/\sqrt{2})$, giving opposite vertex distance $a\sqrt{2}$.