Shortest cut line for cube surface

To obtain a plane scan of the cube, it is necessary to make a cut of the surface. Find the minimum length of the cut line of the cube surface to obtain a flat surface scan. Find minimum length of cut for cube edge equal 1. If the answer can be expressed as $\sqrt{U + V\sqrt{W}},$ where $U,V,W$ are positive integers and $W$ is square-free, give the value of $U+V+W$ .

Solution use approach