Regardless of the size of the square, half the diagonal is (√2)/2 times the side of the square.
The ratio is (√2)/2.
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Consider a square of side length 1. The Pythagorean theorem tells you the diagonal measure (d) is ...
... d² = 1² +1² = 2
... d = √2
The distance from the center of the square to one of its corners (on the circumscribing circle) is then d/2 = (√2)/2. This is the radius of the circle in which our unit square is inscribed.
Since we're only interested in the ratio of the radius to the side length, using a side length of 1 gets us to that ratio directly.
