Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.