Let l="line which connects top corner of cone and one of the points on the corner of base"
Also a="angle between 'l' and surface"

A line is drawn on the surface with distance d to the center of cone. Then two 2d planes are drawn through this line. The angle between plane and surface is equal to 'a'. These planes are directed in the opposite angles. Set of points that are in the cone and between these 2d planes will fall in the slit