In the diagram, AB is an arc of a circle with centre O and radius 4 cm. Angle AOB is \(\alpha\) radians. The point D on OB is such that AD is perpendicular to OB. The arc DC, with centre O, meets OA at C.

(i) Find an expression in terms of \(\alpha\) for the perimeter of the shaded region ABDC.

(ii) For the case where \(\alpha = \frac{1}{6}\pi\), find the area of the shaded region ABDC, giving your answer in the form \(k\pi\), where \(k\) is a constant to be determined.