[ $[$ Dihedral Angle ]

Find 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 equal to $a$.

#
Let $A B C D A 1 B 1 C 1 D 1$ be a regular prism with base $A B C D$ and lateral edges $A A 1, B B 1, C C 1$, and $D D 1$, where $A B=a$. Since the line $A D$ is perpendicular to the plane of the face $C D D 1 C 1$, $D C 1$ is the orthogonal projection of the diagonal $A C 1$ onto this plane, and $A C 1 D$ is the angle between the line $A C 1$ and the plane of the face $C D D 1 C 1$. According to the problem, $\angle A C 1 D=30^{\circ}$. From the right triangles $A C 1 D$ and $D C C 1$, we find that

$$
D C 1=A D \cdot \operatorname{ctg} 30^{\circ}=a \sqrt{3}
$$

$$
C C 1=\sqrt{D C^{2}-C D^{2}}=\sqrt{3 a^{2}-a^{2}}=a \sqrt{2}
$$

since $C C 1$ is the height of the prism, then

$$
V_{A B C D A} 1 B 1 C 1 D 1=S_{A B C D} \cdot C C 1=a 2 \cdot a \sqrt{2}=a 3 \sqrt{2}
$$

## Answer

$a 3 \sqrt{2}$.