1. **State the problem:** Find the area of a right triangle with height $48$ cm and base $\sqrt{108}$ cm. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Substitute the given values:** $$\text{Area} = \frac{1}{2} \times \sqrt{108} \times 48$$ 4. **Simplify $\sqrt{108}$:** $$\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}$$ 5. **Substitute back:** $$\text{Area} = \frac{1}{2} \times 6\sqrt{3} \times 48$$ 6. **Multiply constants:** $$\text{Area} = \frac{1}{2} \times 288\sqrt{3}$$ 7. **Simplify the fraction:** $$\text{Area} = \cancel{\frac{1}{2}} \times \cancel{288}\sqrt{3} = 144\sqrt{3}$$ 8. **Final answer:** $$\boxed{144\sqrt{3} \text{ cm}^2}$$ This is the area of the right triangle.