Question

practice
why is the lateral surface area of a triangular prism always less than the total surface area of a triangular prism? justify your response.

Explanation:

The lateral surface area (LSA) of a triangular prism is the sum of the areas of the three rectangular lateral faces. The total surface area (TSA) is the LSA plus the area of the two triangular bases. Since the area of the two triangular bases is a positive value (as a triangle with non - zero dimensions has a positive area), when we add this positive value to the LSA to get the TSA, the TSA will be greater than the LSA. Mathematically, if \( LSA = \text{sum of lateral faces' areas} \) and \( TSA=LSA + 2\times(\text{area of triangular base}) \), and \( 2\times(\text{area of triangular base})>0 \) (because the base has a non - zero area), then \( TSA>LSA \).

Answer:

The lateral surface area (LSA) of a triangular prism is the area of the three rectangular lateral faces. The total surface area (TSA) is \( \text{LSA}+2\times(\text{area of the triangular base}) \). Since the area of the two triangular bases is a positive quantity (a triangle with non - zero side lengths has a positive area), adding a positive quantity to the LSA (to get TSA) means that \( \text{TSA}>\text{LSA} \), so the lateral surface area of a triangular prism is always less than the total surface area.