1. **Problem statement:** Calculate the surface area of a triangular prism. 2. **Formula:** The surface area $SA$ of a triangular prism is given by: $$SA = ext{Perimeter of triangular base} \times \text{length} + 2 \times \text{Area of triangular base}$$ 3. **Explanation:** - The prism has two triangular bases. - The lateral surface area is the perimeter of the triangle times the length of the prism. - The total surface area is the sum of the lateral area and the areas of the two triangular bases. 4. **Steps to solve:** - Find the perimeter $P$ of the triangular base by adding the lengths of its three sides. - Find the area $A$ of the triangular base using the formula: $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ - Multiply the perimeter $P$ by the length $L$ of the prism to get the lateral surface area. - Multiply the area $A$ by 2 to account for both triangular bases. - Add these two results to get the total surface area: $$SA = P \times L + 2A$$ 5. **Example:** If the triangular base has sides 3, 4, and 5 units, the base is 3 units, height is 4 units, and the prism length is 10 units: - Perimeter $P = 3 + 4 + 5 = 12$ - Area $A = \frac{1}{2} \times 3 \times 4 = 6$ - Lateral area $= 12 \times 10 = 120$ - Total surface area $= 120 + 2 \times 6 = 132$ Thus, the surface area is 132 square units.