The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are A, B, C, D, E and F. Points P and Q are situated where straight sections of the rope meet the pipe with centre A.

(a) Show that angle PAQ = \(\frac{1}{3} \pi\) radians.

(b) Find the length of the rope.

(c) Find the area of the hexagon ABCDEF, giving your answer in terms of \(\sqrt{3}\).

(d) Find the area of the complete region enclosed by the rope.