Answer
The length of the diagonal is $\sqrt{3}~e$
Work Step by Step
The edge of the cube has a length of $e$
Let vertex A be a vertex on the bottom of the cube. Let vertex B be the vertex on the bottom of the cube that is diagonally opposite vertex A.
We can calculate the the length $d$ of a diagonal across the bottom surface of the cube joining vertex A and vertex B:
$d = \sqrt{e^2+e^2} = \sqrt{2}~e$
Let vertex C be a vertex at the top corner of the cube adjacent to vertex B.
The three vertices A,B, and C form a triangle. We need to calculate the length $L$ of the line $\overline{AC}$:
$L = \sqrt{(\overline{AB})^2+(\overline{BC})^2}$
$L = \sqrt{(\sqrt{2}~e)^2+(e)^2}$
$L = \sqrt{2~e^2+e^2}$
$L = \sqrt{3~e^2}$
$L = \sqrt{3}~e$
The length of the diagonal is $\sqrt{3}~e$
