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Two Circles in Square of side 1

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Note that the distance from the lower-left corner of the square to the point where the lower circle touches the sides of the square is exactly the radius of the circle. (The same holds for the upper-right corner and the upper circle, of course.)

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Draw a line connecting the centers of the circle. From each circle center draw a line to the edge of the square horizontally (or vertically, pick one).

So there are 2 horizontal lines equal to the radius of the circle and a 45 degree line connecting the centers of the circle with length twice the radius. The total horizontal distance covered by these lines is the edge of the square. From here, it should be easy to figure out the radius of the circle.
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