3.1. Determine the total surface area of a cube if the distance between non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest integer.
Answer: 1152. Solution. Let's take as the diagonals specified in the condition the diagonals $A_{1} C_{1}$ and $A D_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Construct two parallel planes $A_{1} C_{1} B$ and $A D_{1} C$, each containing one of these diagonals. The distance between these planes is equal to the given distance $d$ between the diagonals. This distance can be expressed in terms of the edge length $a$ of the cube, for example, by considering the section $B_{1} D_{1} D B$ perpendicular to these two planes. The diagonal of the cube $B_{1} D$ is perpendicular to both planes and is divided by them into three equal parts. We obtain: $d=\frac{a \sqrt{3}}{3}$, that is, $a=d \sqrt{3}$. Then $S=6 a^{2}=6 \cdot 3 d^{2}=18 d^{2}$.