11.022. Determine the volume of a regular quadrilateral prism if its diagonal forms an angle of $30^{\circ}$ with the plane of a lateral face, and the side of the base is $a$.
Solution.

Let $A B=B C=C D=D A=a, B_{1} D$ be the diagonal of the prism, $\angle D B_{1} A=30^{\circ}$ (Fig. 11.17). Denote $A B_{1}=b$. Then $b=\frac{a}{\operatorname{tg} 30^{\circ}}=a \sqrt{3}$. The height
$h=B B_{1}=\sqrt{b^{2}-a^{2}}=a \sqrt{2}$. Volume $V=S_{\text {base }} \cdot h$. Since $A B C D$ is a square, then $S_{\text {base }}=a^{2}$. Therefore, $V=a^{2} \cdot a \sqrt{2}=a^{3} \sqrt{2}$.

Answer: $a^{3} \sqrt{2}$.