12.324. In a regular quadrilateral pyramid, a section parallel to the base is made. A line passing through the vertex of the base and the opposite (i.e., not belonging to the same face) vertex of the section forms an angle $\alpha$ with the plane of the base. Find the area of the section if the lateral edge of the pyramid is equal to the diagonal of the base and is equal to $a$.
Solution.

Let $EO$ be the height of the regular pyramid $EABCD$, and let the square $A'B'C'D'$ be a section of the pyramid, with $\angle BDB'=\alpha$ (Fig. 12.185).

Since $BE=DE=BD$, then $\angle EBD=\frac{\pi}{3}, \angle BB'D=\frac{2\pi}{3}-\alpha$.

In $\triangle BB'D: \frac{DB'}{\sin \angle EBD}=\frac{BD}{\sin \angle BB'D} \Rightarrow DB'=\frac{a \sin \frac{\pi}{3}}{\sin \left(\frac{2\pi}{3}-\alpha\right)}=\frac{a \sqrt{3}}{2 \sin \left(\frac{\pi}{3}+\alpha\right)}$.

If $B'F$ is the height of the isosceles trapezoid $BB'D'D$, then from

$$
\Delta B'FD\left(\angle B'FD=\frac{\pi}{2}\right): FD=B'D \cos \angle BDB'=\frac{a \sqrt{3} \cos \alpha}{2 \sin \left(\frac{\pi}{3}+\alpha\right)}
$$

Since $FD=\frac{BD+B'D'}{2}$, then

$$
\begin{aligned}
& B'D'=2FD-BD=\frac{a \sqrt{3} \cos \alpha}{\sin \left(\frac{\pi}{3}+\alpha\right)}-a=\frac{a}{\sin \left(\frac{\pi}{3}+\alpha\right)}\left(\sqrt{3} \cos \alpha-\sin \left(\frac{\pi}{3}+\alpha\right)\right)= \\
& =\frac{a}{\sin \left(\frac{\pi}{3}+\alpha\right)}\left(2 \sin \frac{\pi}{3} \cos \alpha-\sin \frac{\pi}{3} \cos \alpha-\cos \frac{\pi}{3} \sin \alpha\right)=\frac{a \sin \left(\frac{\pi}{3}-\alpha\right)}{\sin \left(\frac{\pi}{3}+\alpha\right)} \\
& \text { The area of the section } S=\frac{1}{2} B'D'^2 \frac{a^2 \sin^2\left(\frac{\pi}{3}-\alpha\right)}{2 \sin^2\left(\frac{\pi}{3}+\alpha\right)} \\
& \text { Answer: } \frac{a^2 \sin^2\left(\frac{\pi}{3}-\alpha\right)}{2 \sin^2\left(\frac{\pi}{3}+\alpha\right)}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1081.jpg?height=664&width=615&top_left_y=80&top_left_x=80)

Fig. 12.186

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1081.jpg?height=434&width=486&top_left_y=288&top_left_x=742)

Fig. 12.186, a