1. **State the problem:** We need to find the surface area of a square-based pyramid with four identical triangular faces. The base side length is 30 mm, and each triangular face has a height of 36 mm and a slant side of 39 mm. 2. **Formula for surface area of a square pyramid:** $$\text{Surface Area} = \text{Base Area} + \text{Lateral Area}$$ where $$\text{Base Area} = s^2$$ and $$\text{Lateral Area} = 4 \times \text{Area of one triangular face}$$ 3. **Calculate the base area:** $$s = 30 \text{ mm}$$ $$\text{Base Area} = 30^2 = 900 \text{ mm}^2$$ 4. **Calculate the area of one triangular face:** The triangle has base $30$ mm and height $36$ mm. $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 30 \times 36$$ $$= 15 \times 36 = 540 \text{ mm}^2$$ 5. **Calculate the total lateral area:** $$4 \times 540 = 2160 \text{ mm}^2$$ 6. **Calculate the total surface area:** $$900 + 2160 = 3060 \text{ mm}^2$$ **Final answer:** $$\boxed{3060 \text{ mm}^2}$$