Let $a$ be the length of the cube's edge, and $\theta$ the cone's half angle. Then $\tan\theta=\sqrt{2}/6$. Consider the vertical cross section of the cone that passes through the apex and a diagonal of the top face, whose length is $a\sqrt{2}$; the height of the cone passes through the apex and the middle point of this diagonal. Let the height from the apex to the top face be $x$; then $6-x=a,$ and $\tan\theta=(a\sqrt{2}/2)/x.$ Therefore $x=3a$ and thus $a=3/2.$