(i) The arc length of AB is given by \(4\alpha\) since the radius is 4 cm and the angle is \(\alpha\) radians.

The arc length of DC is \((4 \cos \alpha) \alpha\) because the radius of the arc DC is \(4 \cos \alpha\).

The length AC (or DB) is \(4 - 4 \cos \alpha\) since AD is perpendicular to OB.

Thus, the perimeter of the shaded region ABDC is \(4\alpha \cos \alpha + 4\alpha + 8 - 8\cos \alpha\).

(ii) For \(\alpha = \frac{1}{6}\pi\), the length OD is \(4 \cos \frac{\pi}{6} = 2\sqrt{3}\).

The area of the sector AOB is \(\frac{1}{2} \times 4^2 \times \frac{\pi}{6}\).

The area of the sector ODC is \(-\frac{1}{2} (2\sqrt{3})^2 \times \frac{\pi}{6}\).

The shaded area is \(\frac{\pi}{3}\), which can be expressed as \(k\pi\) where \(k = \frac{1}{3}\).