1. **State the problem:** We need to find the area of a right triangle with base $\sqrt{108}$ cm and height $4.8$ cm. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Calculate the base:** $$\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}$$ 4. **Substitute values into the area formula:** $$\text{Area} = \frac{1}{2} \times 6\sqrt{3} \times 4.8$$ 5. **Simplify the multiplication:** $$= \frac{1}{2} \times 6 \times 4.8 \times \sqrt{3} = \frac{1}{2} \times 28.8 \times \sqrt{3}$$ 6. **Simplify further:** $$= 14.4 \times \sqrt{3}$$ 7. **Approximate $\sqrt{3} \approx 1.732$:** $$\text{Area} \approx 14.4 \times 1.732 = 24.9408$$ 8. **Final answer:** The area of the triangle is approximately $24.94$ cm$^2$, not $36$ cm$^2$.