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This is an interesting question: very simple but incredibly difficult. I have not found a proper answer but have got a partial answer to a related question: given a number of unit squares, what is the smallest circle that they will fit into. The two are linked very simply: if 2 unit squares will fit in a circle of radius 1.118 then a unit circle can hold 2 squares of a maximum side of 1/1.118 units.
The answers given below can be proved only for n = 1 and 2.
1 square : radius = sqrt(2)/2 = 0.707 approx.
2 squares : radius = sqrt(5)/2 = 1.118 approx.
3 squares : radius = 1.288
4 squares : radius = 1.414
5 squares : radius = 1.581
10 squares : radius = 2.121
20 squares : radius = 2.893
30 squares : radius = 3.485.
For all integers up to 35 see www2.stetson.edu/~efriedma/squincir/
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