1. **State the problem:** We need to find the surface area of a triangular prism with triangular base sides 13 cm, 16 cm, and 12 cm, and prism length (depth) 5 cm. 2. **Calculate the area of the triangular base:** Use Heron's formula. First, find the semi-perimeter: $$s = \frac{13 + 16 + 12}{2} = \frac{41}{2} = 20.5$$ 3. Calculate the area $A$ of the triangle: $$A = \sqrt{s(s - 13)(s - 16)(s - 12)} = \sqrt{20.5(20.5 - 13)(20.5 - 16)(20.5 - 12)}$$ $$= \sqrt{20.5 \times 7.5 \times 4.5 \times 8.5}$$ Calculate the product inside the square root: $$20.5 \times 7.5 = 153.75$$ $$4.5 \times 8.5 = 38.25$$ $$153.75 \times 38.25 = 5880.9375$$ So, $$A = \sqrt{5880.9375} \approx 76.68 \text{ cm}^2$$ 4. **Calculate the perimeter of the triangular base:** $$P = 13 + 16 + 12 = 41 \text{ cm}$$ 5. **Calculate the lateral surface area:** This is the perimeter times the length of the prism: $$L = P \times 5 = 41 \times 5 = 205 \text{ cm}^2$$ 6. **Calculate total surface area:** The prism has two triangular bases and three rectangular faces: $$\text{Surface Area} = 2 \times A + L = 2 \times 76.68 + 205 = 153.36 + 205 = 358.36 \text{ cm}^2$$ **Final answer:** The surface area of the triangular prism is approximately **358.36 cm\textsuperscript{2}**.