Question 125148
The shortest path around the outside of the cube is perpendicular to the nearest edge of the cube, straight across the next face through its center, and then perpendicular to the edge of the cube to the next point.


The first part of the path is exactly half of the length of an edge, the second part is exactly the length of an edge, and the third part is again half the length of an edge.  So the entire path is 2 times the length of an edge.  But we are given that the length of the path is {{{2(sqrt(2))}}}, therefore the length of one edge is {{{sqrt(2)}}}.


The volume is the length of an edge cubed, so {{{V=(sqrt(2))^3=red(2*sqrt(2))}}}