Click here for GSP file of this figure.

We want to find

We can construct a triangle from the midpoint of the base of the square to one of the vertices of the square. This triangle is outlined in pink and the lengths of the sides are labeled.

Now we can use Pythagorean Theorem to get

Using some algebra we can solve for r

Now we have two expressions for the radius of the circle. We can set these equations equal to each other and simplify to get

Finally, we can rationalize to get

Which is the famous Golden Ratio!! (Click on the link for more info about the Golden Number)

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