How do you find inner diagonals of a cube? I was wondering that for a very long time.
Let a be the length of every side. D'B is an inner diagonal of the cube ABCDA'B'C'D'. B'D, A'C, C'A are also inner diagonals.
DD' $\displaystyle \perp$ (ABCD), so DD' is perpendicular to DB (since DB is in (ABCD)), that means that BDD' is a right triangle:
$\displaystyle D'D^2+DB^2=D'B^2 \Rightarrow D'B=\sqrt{D'D^2+DB^2}$
ABD is a right triangle, so: $\displaystyle AD^2+AB^2=DB^2 \Rightarrow DB^2=2a^2$
$\displaystyle D'B=\sqrt{a^2+2a^2}=\sqrt{3a^2}=a\sqrt3$