You can use trig to get the same answer as those above.
Draw three lines: One from the center of the circle to the corner shared by the square and the rectangle. Then draw a line from the center of the circle to the corner nearest to the center of the circle. Draw a final line being the diagonal connecting the previously mentioned corners.
We know the length of the third line by the pythagorean theorem. If we call the side length of the square L, the length of the shorter of the two remaining lines is L/2. The length of the longer, L/sqrt(2).
Find the angle that the diagonal makes with the longer of the drawn lines allows you to apply the cosine rule.
The longer line meets the square's corner at a 45 degree angle with respect to either side. Then angle the diagonal makes with the left side of the square has a tangent of 2.
Apply the cosine rule then solve the resulting quadratic and you get two possible answers, only one of which is plausible.