WHAT!!!! you must be joking

Circle in a chess board

Consider an infinite chessboard, where the squares have side length of 1. The squares are colored black and white alternately. The (finite) radius of the largest circle whose circumference can be drawn completely on the white squares (hence you can see the entire circle) has the form , where  and  are integers,  and  are coprime, and  is not divisible by the square of any prime. What is the value of ?

Details and assumptions

 and  are all allowed to be 1. In particular, if you think the the largest radius is , then your answer to this should be .
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