**1-1/sqrt(2)**

See picture for details.

x- the required value

r- is the radius of sphere=1

it is obvious the distance AO=sqrt(2) as sides of the large square = r=1

AC=AO-r=sqrt(2)-1

AC=x*sqrt(2) => x=AC/sqrt(2)=(sqrt(2)-1)/sqrt(2)=1-1/sqrt(2)

x=1-1/sqrt(2)