(i) To find the length of the arc BD, we first calculate the length of AB using the Pythagorean theorem:

\(AB = \sqrt{6^2 + 6^2} = \sqrt{72}\)

The angle BAD is 45° or \(\frac{\pi}{4}\) radians. The arc length s is given by \(s = r\theta\), where \(r = \sqrt{72}\) and \(\theta = \frac{\pi}{4}\):

\(s = \sqrt{72} \times \frac{\pi}{4} \approx 6.67 \text{ cm}\)

(ii) To find the area of the shaded region, we calculate the area of the sector and subtract the area of triangle OAB. The sector area is given by:

\(\text{Sector area} = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 72 \times \frac{\pi}{4}\)

The area of triangle OAB is:

\(\text{Area of triangle} = \frac{1}{2} \times 6 \times 6 = 18 \text{ cm}^2\)

Thus, the shaded area is:

\(\text{Shaded area} = \frac{1}{2} \times 72 \times \frac{\pi}{4} - 18 \approx 10.3 \text{ cm}^2 \text{ or } 9\pi - 18 \text{ cm}^2\)