Circumscribed Stacked Squares

The circle of smallest radius will pass through the top two vertices of the square on top, and also the bottom rightmost vertex and the bottom leftmost vertex of the bottom squares. This circle's center will be on the common side of the two bottom squares. Let $x$ denote the distance from the center of the circle to the bottom vertex of the common side of the two bottom squares. Then $16 - x$ is the distance from the center of the circle to the top vertex of the common side of the two bottom squares, and $32-x$ is the distance from the center to the top of the top square. Let $r$ denote the radius of the circle.

By the Pythagorean Theorem, $r^2 = 8^2 + (32-x)^2$ , and also $r^2 = 16^2 + x^2$ . Therefore, $8^2 + (32-x)^2 = 16^2 + x^2$ , and when we simplify this equation the $x^2$ terms cancel and we get $832 = 64x$ . Therefore $x = 13$ and $r^2 = 16^2 + 13^2 = 425$ .