Determine the length of a side of an octahedron from the volume

Question

I am looking for a formula that can convert the volume of an octahedron to the length of an edge. So far, I have come across $\frac{1.442\cdot3\sqrt{v}}{1.122}$, but I am not sure if this equation is accurate. Thanks in advance!

Answer 1

See, that octahedron is dual to cube.

The volume of octahedron inscribed in cube is equal to $\frac{1}{6}$ volume of cube, ie: $$v_o = \frac{1}{6}v_c$$ And the length of a side of octahedron is equal to $\frac{\sqrt{2}}{2}$ side of cube, ie: $$l_o = \frac{\sqrt{2}}{2}l_c$$ Length of side of cube is equal to cubic root of it's volume: $$l_c = \sqrt[3]{v_c}$$

Then we have: $$l_o = \frac{\sqrt{2}}{2}\sqrt[3]{6v_o}$$