The diagram shows a circle, centre \(O\) of radius \(r\text{ cm}\), and a chord \(AB\). Angle \(AOB=\theta\) radians.

The length of the major arc \(AB\) is \(5\) times the length of the minor arc \(AB\). The minor arc \(AB\) has length \(2\pi\text{ cm}\).

(i) Find the value of \(\theta\) and of \(r\).

(ii) Calculate the exact perimeter of the shaded segment.

(iii) Calculate the exact area of the shaded segment.