(i) To find angle DOE, use the sine rule in triangle DOE:

\(\sin \frac{1}{2} \theta = \frac{6}{10}\)

Solving for \(\theta\), we find:

\(\theta = 1.287 \text{ radians}\)

(ii) To find the perimeter of the metal plate, calculate the arc length and add the straight edges:

Arc length of BOD = \(2 \times 10 \times (\pi - 1.287)\)

Perimeter = \(12 + 12 + 2 \times 10 \times (\pi - 1.287) = 61.1 \text{ cm}\)

(iii) To find the area of the metal plate, calculate the area of the sectors and triangles:

Area of sector DOE = \(\frac{1}{2} \times 10^2 \times 1.287\)

Area of triangle DOE = \(\frac{1}{2} \times 10^2 \times \sin 1.287\)

Total area = \(\pi \times 10^2 - (2 \times \text{sectors} - 2 \times \text{triangles})\)

Area = 281 or 282 cm².