Explanation:

- A rhombus is a quadrilateral with all sides equal and opposite angles equal.

- In rhombus OABC, vertices O, A, and B lie on a circle with the center at O.

- Since O is the center and a vertex, the radius is the distance from O to any vertex on the circle.

- The rhombus's area is given as \( 32\sqrt{3} \) square cm.

- The formula for the area of a rhombus is \(\frac{1}{2} \times \text{product of diagonals}\).

- Since O is the center, each diagonal is the diameter of the circle.

- Calculate using \( \frac{1}{2} \times 2r \times 2r = 32\sqrt{3} \) to find the circle's radius \( r \).

- Solving, \( r^2 \sqrt{3} = 32\sqrt{3} \), gives \( r^2 = 32 \).

- Therefore, r = 8 cm.

- From the given options:

- Option 1: 4 cm

- Option 2: 6 cm

- Option 3: 8 cm \(\huge\checkmark\)

- Option 4: 16 cm

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