(i) The area of circle C is \(\frac{1}{2} \times 3^2 \pi = \frac{9}{2} \pi\).

The area of the sector ORS is \(\frac{1}{2} \times 9^2 \theta - \frac{1}{2} \times 3^2 \theta\).

Equating the areas: \(\frac{9}{2} \pi = \frac{1}{2} \times 9^2 \theta - \frac{1}{2} \times 3^2 \theta\).

Simplifying gives \(\theta = \frac{1}{4}\pi\).

(ii) The perimeter of the shaded region includes the arc RS and the straight lines PS and PR.

The arc length RS is \(9 \times \frac{1}{4}\pi\).

The straight lines PS and PR are both 6 cm.

Thus, the perimeter is \(6 + 6 + 9 \times \frac{1}{4}\pi = 12 + 3\pi\) or approximately \(21.4 \text{ cm}\).