[ Basic properties and definitions of regular polyhedra ] Volume of a tetrahedron and a pyramid

Find the volume of a regular octahedron (regular octahedron), with edge length $a$.

#
Consider a regular octahedron as the union of two regular square pyramids $P A B C D$ and $Q A B C D$ with a common base $A B C D$, where the vertices $P$ and $Q$ are located on opposite sides of the plane of the square $A B C D$, and all edges of these pyramids are equal to $a$. Let $O$ be the center of the square $A B C D$. Then $P O$ is the height of the pyramid $P A B C D$. From the right triangle $A O P$, by the Pythagorean theorem, we find that

$$
P O=\sqrt{P A^{2}-O A^{2}}=\sqrt{a^{2}-\left(\frac{a \sqrt{2}}{2}\right)^{2}}=\frac{a \sqrt{2}}{2}
$$

Let $V$ be the volume of the octahedron. Then

$$
V=2 V_{P A B C D}=2 \cdot \frac{1}{3} a^2 \cdot \frac{a \sqrt{2}}{2}=\frac{a^{3} \sqrt{2}}{3}
$$

## Answer

$\frac{a^{3} \sqrt{2}}{3}$