I assume that when you say a circle of 2r, that is the diameter of the circle and not the radius. Proceeding with that assumption, a square with diagonal 2r fits perfectly in a circle of diameter 2r. If the diagonal of a square is 2r, you can use the Pythagorean theorem to find the side of the square. x^2 + x^2 = (2r)^2 . Solving this for x you'd get 2x^2=4r^2 so x^2 =2r^2 and x = r*sqrt(2) So if you put squares of this size next to eachother, the circles would cover all the points included inside those squares. So for a rectangle of let's say 5r by 8r, you just have to calculate how many of these squares fit into the rectangle. So 5r/r sqrt 2 = 5 sqrt 2 /2 which is approximately 3.5,so you'll need 4 squares in that direction by 8r/r sqrt 2 which is 8/sqrt 2 which is approximately 5.7, so you'd need 6 squares. So for this specific case you'd need 6*4 or 24 squares which means 24 circles. Hope this helps you out.

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