Circles and Squares

circles and squares 1

Here are three circles and two squares, inscribed successively as shown.

If the diameter of the largest circle is 10, what is the diameter of the smallest circle?

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There are a number of ways to solve this. The simplest is to rotate the smaller square 45 degrees: Now the radius of the largest circle (red), which we know to be 5, equals the side of the smaller square (green), because they’re the two diagonals of a square. So the diameter of the smallest circle is 5.