For a square with side length a, the circle inscribed in it has radius a/2, while the circle circumscribed about it (passing through all four vertices) has radius equal to half the square's diagonal, a*sqrt(2)/2; the ratio of their areas is the square of the ratio of their radii.
Inner (inscribed) circle radius r = a/2. Outer (circumscribed) circle radius R = a*sqrt(2)/2. Ratio of areas = R^2/r^2 = [(a*sqrt(2)/2)^2] / [(a/2)^2] = (a^2...
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