1. **State the problem:** We need to find the surface area of a square-based pyramid with a square base side length of 30 mm and four identical triangular faces each having a height of 36 mm and a slant edge 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 area of a triangle is $$\frac{1}{2} \times \text{base} \times \text{height}$$ Here, the base of the triangle is the side of the square base, 30 mm, and the height is given as 36 mm. $$\text{Area} = \frac{1}{2} \times 30 \times 36 = 15 \times 36 = 540 \text{ mm}^2$$ 5. **Calculate the lateral area:** $$4 \times 540 = 2160 \text{ mm}^2$$ 6. **Calculate total surface area:** $$900 + 2160 = 3060 \text{ mm}^2$$ 7. **Final answer:** The surface area of the pyramid is $$\boxed{3060 \text{ mm}^2}$$