Explanation:

To find the volume of a right pyramid with a square base:

- Base Properties: The diagonal of the square base is \(6\sqrt{2}\) cm.

- Side Length Calculation: The diagonal of a square is \(s\sqrt{2}\), where \(s\) is the side length. Thus, \(s\sqrt{2} = 6\sqrt{2}\) implies \(s = 6\).

- Base Area: The area of the square (base) is \(s^2 = 6^2 = 36 \, \text{cm}^2\).

- Pyramid Volume Calculation: The volume \(V\) of a pyramid is given by \(\frac{1}{3} \times \text{Base Area} \times \text{Height}\).

- Compute Volume: \(V = \frac{1}{3} \times 36 \times 12 = 144 \, \text{cm}^3\).

- Options Consideration:

- Option 1: 864 cm\(^3\) is incorrect.

- Option 2: 432 cm\(^3\) is incorrect.

- Option 3: 144 cm\(^3\)

- Option 4: 288 cm\(^3\) is incorrect.

Answer: Option 3: 144 cm\(^3\)