Answer:
(√2)/2
Explanation:
The ratio of the radius of the circle to the side of the inscribed square is the same regardless of the size of the objects.
The radius of the circle is half the length of the diagonal of the square. For simplicity, we can call the side of the square 1, so its diagonal is √(1²+1²) = √2 by the Pythagorean theorem. The radius is half that value, so is (√2)/2. The desired ratio is this value divided by 1.
Scaling up our unit square to one with a side length of 3 inches, we have ...
radius/side = ((3√2)/2) / 3 = (√2)/2
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A square with a side length of 3 inches will have an area of (3 in)² = 9 in².