1. **State the problem:** Find the total surface area of the given triangular prism with triangular base sides 13 cm, 5 cm, and 16 cm, height 12 cm, and length (depth) 30 cm. 2. **Calculate the area of the triangular base:** - Given the base has sides 13 cm, 5 cm, and 16 cm, use Heron's formula. - Calculate semi-perimeter $s=\frac{13+5+16}{2}=\frac{34}{2}=17$ cm. - Area $$A=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{17(17-13)(17-5)(17-16)}=\sqrt{17\times4\times12\times1}=\sqrt{816}$$ - Simplify $$\sqrt{816}=2\sqrt{204}\approx 28.57\text{ cm}^2$$. 3. **Calculate the lateral surface area:** - The perimeter of the triangular base $$P=13+5+16=34\text{ cm}$$. - Multiply by the length of the prism to find lateral surface area: $$34 \times 30 = 1020\text{ cm}^2$$. 4. **Calculate total surface area:** - Total surface area includes two triangular bases plus the lateral faces. - $$\text{Total surface area} = 2 \times 28.57 + 1020 = 57.14 + 1020 = 1077.14\text{ cm}^2$$. **Final answer:** The total surface area of the triangular prism is approximately $1077.14$ cm².