11.052. In a regular triangular prism, the area of the section passing through a lateral edge perpendicular to the opposite lateral face is $Q$. The side of the base of the prism is $a$. Find the total surface area of the prism.
Solution.

$$
S=2 S_{\mathrm{oCH}}+3 S_{B B_{1} A_{i} A} \text { (Fig. 11.47); } C D=\frac{a \sqrt{3}}{2} ; A B=B C=C A=a \text {; }
$$

$$
\begin{aligned}
& Q=C D \cdot C_{1} C=\frac{a \sqrt{3}}{2} \cdot C_{1} C ; C_{1} C=\frac{2 Q}{a \sqrt{3}} ; S_{\mathrm{och}}=\frac{a^{2} \sqrt{3}}{4} ; \\
& S_{B B_{1} A_{1} A}=A B \cdot B B_{1}=A B \cdot C_{1} C=\frac{2 Q}{a \sqrt{3}} \cdot a=\frac{2 Q}{\sqrt{3}} ; \\
& S=\frac{a^{2} \sqrt{3}}{2}+2 Q \sqrt{3}=\sqrt{3}\left(0.5 a^{2}+2 Q\right) . \\
& \text { Answer: } \quad \sqrt{3}\left(0.5 a^{2}+2 Q\right)
\end{aligned}
$$