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.
The ratio of the radius of the circle to the side of the square is [tex]\dfrac{1}{\sqrt 2}.[/tex]
A ratio shows us the number of times a number contains another number.
The area of the square is equal to the square of its side, therefore, the length of the side of the square can be written as,
[tex]\text{Area of square} = a^2\\\\9 = a^2\\\\a = 3\rm\ inches[/tex]
As we know that the diagonal of the square that is inscribed in a circle is the diameter of that circle. Therefore, the radius of the circle can be written as,
[tex]\rm Radius = \dfrac{\text{Diameter of circle}}{2} =\dfrac{\text{Diagonal of the square}}{2}= \dfrac{a\sqrt 2}{2}[/tex]
[tex]\rm Radius = \dfrac{a\sqrt 2}{2}\\\\\rm Radius = \dfrac{3\sqrt 2}{2}[/tex]
Thus, the radius of the circle is [tex]\dfrac{3\sqrt 2}{2}[/tex] inches.
Now, the ratio of the radius of the circle to the side of the square can be written as,
[tex]\dfrac{\text{Radius of the circle}}{\text{Side of the square}} = \dfrac{\frac{3\sqrt 2}{2}}{3} = \dfrac{3\sqrt 2}{2 \times 3} = \dfrac{\sqrt 2}{2} = \dfrac{1}{\sqrt 2}[/tex]
Hence, the ratio of the radius of the circle to the side of the square is [tex]\dfrac{1}{\sqrt 2}.[/tex]
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