Let us say that the square have edge = sqrt(pi), then its area is pi since A = edge^2, and therefore the diagonal is pi*sqrt(2).
Meanwhile the circle of radius 1 has area pi since area of circle is A = pi *radius^2.
Therefore we get that the diagonal of the square divided by the radius is:
diagonal / radius = pi*sqrt(2) / 1 = pi*sqrt(2)