Asquare is inscribed in a circle. if the area of the square is 9 in2, what is the ratio of the radius of the circle to the side of the square?

Regardless of the size of the square, half the diagonal is (√2)/2 times the side of the square.

The ratio is (√2)/2.

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.

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