11.036. Determine the volume of an octahedron (a regular octahedron), the edge of which is equal to $a$.
Solution.

$$
V=\frac{1}{3} S H, \text { where } H=2 \cdot h=2 \cdot S O, S=a^{2}
$$

(Fig. 11.31). In $\triangle S O C \angle S O C=90^{\circ}$, then $S O^{2}=S C^{2}-O C^{2}$, where $O C=\frac{a \sqrt{2}}{2}$; $S O^{2}=\frac{2 a^{2}-a^{2}}{2}=\frac{a^{2}}{2} ; h=\frac{a}{\sqrt{2}}, H=\frac{2 a}{\sqrt{2}}$, and thus,

$$
\begin{aligned}
& V=\frac{1}{3} S H=\frac{1}{3} a^{2} \cdot \frac{2 a}{\sqrt{2}}=\frac{\sqrt{2}}{3} a^{3} \\
& \text { Answer: } \frac{a^{3} \sqrt{2}}{3}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-672.jpg?height=822&width=484&top_left_y=901&top_left_x=764)

Fig. 11.31

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-673.jpg?height=438&width=472&top_left_y=93&top_left_x=124)

Fig. 11.32

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-673.jpg?height=430&width=618&top_left_y=99&top_left_x=617)

Fig. 11.33