Unit sphere is on a flat plane in rain. What is the length of largest cube that can hide underneath it and not get wet?

Question

A unit sphere (radius = 1) is out on a flat plane in the rain. Find the side length of the largest cube that can hide underneath it and not get wet.

Answer 1

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)

---