In the diagram, the vertical blue line is equal to the radius of the circle O. Since the triangle formed by the red line, the vertical blue line, and the base of the square is a 45-45-90 triangle, this means the length of the red line is r.

Since the radius of circle C is 2, the length of the diagonal line, from the corner to of the square to the center of C, is

r + r + 2

But this is half the diagonal of the square, or 2.

Thus r + r + 2 = 2

r( +1) = 2 - 2

r=

Simplifying this gives us r= 6 - 4

2