Question 1019491
Let's first solve for the largest square that can be cut from a circle of radius r.
The diameter of the circle becomes the diagonal of the square so you can solve for the side of the square since the diagonal is (using the Pythagorean theorem),
{{{D[s]^2=s^2+s^2}}}
{{{D[s]^2=2s^2}}}
{{{D[s]=sqrt(2)*s}}}
{{{2r=sqrt(2)*s}}}
{{{s=(2r)/sqrt(2)}}}
{{{s^2=(4r^2)/2=2r^2}}}
So to make a cube, use that same distance s. 
So a diagonal for a cube of side s would be,
{{{D[c]=sqrt(s^2+s^2+s^2)}}}
{{{D[c]=sqrt(3)*s}}}
{{{D[c]=sqrt(3)((2r)/sqrt(2))}}}
{{{D[c]=sqrt(6)*r}}}