Octahedron Volume

What is the volume of a regular octahedron?

octahedron
Solution

The volume is given by $V = \frac{\sqrt{2}}{3}$

To arrive at this, first note that the octahedron is two square based pyramids, sharing the square face. The volume of a pyramid is $V_P = \frac{1}{3}bh$. The volume of the octahedron is $V_{Oct} = 2V_P$. The perpendicular height of the pyramids is $\frac{1}{\sqrt{2}}$, giving $V_{Oct} = 2\times \frac{1}{3}\times 1 \times \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{3}$