1. **State the problem:** We need to find the surface area of a square-based pyramid with a base side length of 9 cm and four identical triangular faces, each with a height of 14 cm. 2. **Formula for surface area of a square-based 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:** $$s = 9 \text{ cm}$$ $$\text{Base Area} = 9^2 = 81 \text{ cm}^2$$ 4. **Calculate the area of one triangular face:** The area of a triangle is given by $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ Here, the base of the triangle is the side of the square base, 9 cm, and the height of the triangle is 14 cm. $$\text{Area of one triangle} = \frac{1}{2} \times 9 \times 14 = \frac{1}{2} \times 126 = 63 \text{ cm}^2$$ 5. **Calculate the lateral area:** $$\text{Lateral Area} = 4 \times 63 = 252 \text{ cm}^2$$ 6. **Calculate the total surface area:** $$\text{Surface Area} = 81 + 252 = 333 \text{ cm}^2$$ **Final answer:** The surface area of the pyramid is $$\boxed{333 \text{ cm}^2}$$