11.037. The base of the prism is a square with a side equal to $a$. One of the lateral faces is also a square, the other is a rhombus with an angle of $60^{\circ}$. Determine the total surface area of the prism.
Solution.

$S_{\text {full }}=4 \cdot S_{A B C D}+2 \cdot S_{B C C_{1} B_{1}}$ (Fig. 11.32), $S_{A B C D}=a^{2}$ (since $A B C D-$ is a square and $A B=a), \quad S_{B C C_{1} B_{1}}=\frac{1}{2} a^{2} \sin 60^{\circ}=\frac{a^{2} \sqrt{3}}{2}\left(B B_{1} C_{1} C-\right.$ is a rhombus, $\left.B B_{1}=a, \angle B_{1} C_{1} C=60^{\circ}\right), S_{\text {full }}=4 a^{2}+a^{2} \sqrt{3}=a^{2}(4+\sqrt{3})$.

Answer: $\quad a^{2}(4+\sqrt{3})$.