12.129. Find the volume of a regular quadrilateral pyramid if the side of its base is equal to $a$, and the dihedral angle at the base is $\alpha$.
Solution.

The volume of the pyramid $V=\frac{1}{3} S H$ (Fig. 12.129). The area of the base: $S=a^{2}$. The height of the pyramid from $\triangle S O E \quad S O=H=\frac{a}{2} \operatorname{tg} \alpha$. Therefore, $V=\frac{1}{3} \frac{a^{3}}{2} \operatorname{tg} \alpha=\frac{a^{3}}{6} \operatorname{tg} \alpha$.

Answer: $\frac{a^{3}}{6} \operatorname{tg} \alpha$.