3.29. The side of the base of a regular quadrilateral prism is equal to $a$. The angle between the intersecting diagonals of two adjacent lateral faces is $\alpha$. Find the volume of the prism.
3.29. According to the condition, in the regular quadrilateral prism \(ABCD A_1 B_1 C_1 D_1\), we have \(AB = a\), \(\angle A_1 DC_1 = \alpha\) (Fig. 3.31). Since \(A_1 C_1\) is the diagonal of the square, then \(A_1 C_1 = a \sqrt{2}\). Let \(A_1 D = x\); then \(DC_1 = x\). From \(\Delta A_1 DC_1\) by the cosine theorem, we find

\[
2a^2 = 2x^2 - 2x^2 \cos \alpha, \text{ or } 2a^2 = 4x^2 \sin^2 \frac{\alpha}{2},
\]

from which \(x = \frac{a \sqrt{2}}{2 \sin \frac{\alpha}{2}}\). Further, from \(\triangle DD_1 A_1\) we get

\[
DD_1 = \sqrt{\frac{2a^2}{4 \sin^2 \frac{\alpha}{2}} - a^2} = \frac{a \sqrt{2(1 - 2 \sin^2 \frac{\alpha}{2})}}{2 \sin \frac{\alpha}{2}} = \frac{a \sqrt{2 \cos \alpha}}{2 \sin \frac{\alpha}{2}}
\]

Thus, \(V_{\text{pr}} = \frac{a^3 \sqrt{2 \cos \alpha}}{2 \sin \frac{\alpha}{2}}\).

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-517.jpg?height=519&width=421&top_left_y=1383&top_left_x=43)

Fig. 3.32