How critical is it to get the maximum number of rectangles inside the circle?
This is starting to look like a packing problem. There are too many variables (unknowns) for a unique solution. Each size rectangle will have an area that CANNOT be covered -- you want to minimize that "loss".
My first attempt would be to ONLY use the first quadrant (then multipy by 4).
Start at the origin and place rectangles along the x-axis until it touches the edge of the circle.
Stack the next set of rectangle atop the first set.
At some point you will have an area left over that MAY allow the placement of more rectangles.
For specific rectangle dimensions, there is a maximum you can fit inside the circle.