1. **State the problem:** We need to find the surface area of a square-based pyramid with a square base edge of 8.6 cm and four identical isosceles triangular faces, each with a slant height of 12 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 $$\text{Lateral Area} = \frac{1}{2} \times \text{Perimeter of base} \times \text{Slant height}$$ 3. **Calculate the base area:** $$s = 8.6 \text{ cm}$$ $$\text{Base Area} = 8.6^2 = 73.96 \text{ cm}^2$$ 4. **Calculate the perimeter of the base:** $$\text{Perimeter} = 4 \times 8.6 = 34.4 \text{ cm}$$ 5. **Calculate the lateral area:** $$\text{Lateral Area} = \frac{1}{2} \times 34.4 \times 12 = 0.5 \times 34.4 \times 12$$ $$= 0.5 \times 412.8 = 206.4 \text{ cm}^2$$ 6. **Calculate total surface area:** $$\text{Surface Area} = 73.96 + 206.4 = 280.36 \text{ cm}^2$$ 7. **Round to nearest cm²:** $$\boxed{280 \text{ cm}^2}$$ This is the total surface area of the pyramid rounded to the nearest square centimeter.