The square’s diagonal equals the circle’s diameter 2r. Area of the square is (1/2) x diagonal² = (1/2) x (2r)² = 2r². Dividing by circle area πr² gives the ratio 2 / π ≈ 0.637.
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Let shorter diagonal be d, so longer diagonal is 2d. Area of rhombus is (1/2) x d1 x d2 = (1/2) x d x 2d = d². Equating d² = 128 gives shorter diagonal d = √128 = 8√2 cm ...