Question

A circle and a square can intersect at how many points

Answer 1

If the circle has the same center as the diagonals of a square and the radius of the circle is smaller than 1/2 the diagonal of the square but larger than 1/2 the length of the side of a square, then there are 8 points of intersection -- 2 at each corner of the square.

If the radius of the circle is smaller than 1/2 the side length of the square and the center is as described above, there are no points of intersection.

If the circle is located outside the square it can have 1 tangent point or 2 intersection points depending on the location conditions of the circle in relation to the square.