The figure is a regular hexagon ABCDEF (in order) with side '2a' cm, drawn with C at the top vertex, F at the bottom vertex, B and A forming the left vertical edge, and D and E forming the right vertical edge. A rectangle is drawn using side AB as its left edge, with its bottom edge AG horizontal (G lying on line AE) and its right edge GH vertical (parallel to ED, with H directly above G). Given AG = FG and ED || GH, what is the ratio of the area of the shaded region (the triangle below AG down to F, plus the strip to the right of GH up to the hexagon's right edge DE) to the area of the hexagon?