2.17. The base of a right prism is a rhombus. The areas of the diagonal sections of this prism are $P$ and $Q$. Find the lateral surface area of the prism.
2.17. Let the height of the prism be $h$. Since $A B C D$ is a rhombus, then $S_{\text {side }}=4 A B \cdot h$ (Fig. 2.16). Further, $A C \perp B D$ (as diagonals of a rhombus) and,

therefore, $\quad A B=\sqrt{\frac{B D^{2}}{4}+\frac{A C^{2}}{4}}=\frac{\sqrt{B D^{2}+A C^{2}}}{2}$. Then $S_{\text {side }}=2 \sqrt{B D^{2}+A C^{2}} \cdot h=2 \sqrt{B D^{2} \cdot h^{2}+A C^{2} \cdot h^{2}}$. But $B D \cdot h=P$, $A C \cdot h==Q$ (by condition). Thus, $S_{\text {side }}=2 \sqrt{P^{2}+Q^{2}}$.

Answer: $2 \sqrt{P^{2}+Q^{2}}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-464.jpg?height=365&width=515&top_left_y=721&top_left_x=350)

Fig. 2.16