Triangular Prism Area 1336Bc

1. **State the problem:** We need to find the total surface area of a triangular prism with given side lengths. 2. **Identify the dimensions:** The prism has two triangular bases and three rectangular faces connecting corresponding sides. 3. **Calculate the area of the triangular bases:** The triangles are right-angled with legs given. - For the first triangle, legs are $16.9$ cm and $18.2$ cm. - Area of triangle = $\frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 = \frac{1}{2} \times 16.9 \times 18.2$ - Calculate: $$\frac{1}{2} \times 16.9 \times 18.2 = 0.5 \times 16.9 \times 18.2 = 153.79 \text{ cm}^2$$ 4. **Calculate the area of the second triangular base:** Legs are $15.6$ cm and $6.5$ cm. - Area = $\frac{1}{2} \times 15.6 \times 6.5 = 0.5 \times 15.6 \times 6.5 = 50.7 \text{ cm}^2$ 5. **Calculate the lateral faces areas:** These are rectangles formed by connecting corresponding sides. - The prism height (length) is the distance between the triangular bases. Since the problem does not specify, we assume the prism length is the same as the side connecting the triangles, which is $18.2$ cm (or the average if needed). However, the problem likely implies the prism length is $18.2$ cm. - Rectangular faces areas: a) Side corresponding to $16.9$ cm: Area = $16.9 \times 18.2 = 307.58$ cm$^2$ b) Side corresponding to $18.2$ cm: Area = $18.2 \times 18.2 = 331.24$ cm$^2$ c) Side corresponding to $6.5$ cm or $15.6$ cm? Since the triangles differ, we take the average or consider the prism length as the height. The problem is ambiguous, but typically the prism length is the distance between the triangular bases, which is not given explicitly. We assume the prism length is $18.2$ cm. For the third side, take the average of the two corresponding sides: $\frac{6.5 + 15.6}{2} = 11.05$ cm. Area = $11.05 \times 18.2 = 201.01$ cm$^2$ 6. **Sum all areas:** - Total surface area = sum of two triangular bases + sum of three rectangular faces - $153.79 + 50.7 + 307.58 + 331.24 + 201.01 = 1044.32$ cm$^2$ 7. **Final answer:** $$\boxed{1044.32 \text{ cm}^2}$$