Triangular Prism Area 7Df721

1. **State the problem:** We need to find the total surface area of a triangular prism given its net with dimensions 3.3 cm, 4.4 cm, 5.5 cm, and 7.1 cm. 2. **Identify the shapes in the net:** The net consists of two congruent triangles and three rectangles. 3. **Calculate the area of the triangular faces:** The triangle has sides 3.3 cm, 4.4 cm, and 5.5 cm. Use Heron's formula for the area of a triangle: $$s = \frac{3.3 + 4.4 + 5.5}{2} = \frac{13.2}{2} = 6.6$$ $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{6.6(6.6 - 3.3)(6.6 - 4.4)(6.6 - 5.5)}$$ $$= \sqrt{6.6 \times 3.3 \times 2.2 \times 1.1} = \sqrt{52.70292} \approx 7.26 \text{ cm}^2$$ 4. **Calculate the area of the rectangular faces:** The rectangles correspond to the sides of the triangle times the length of the prism (7.1 cm). Areas: $$3.3 \times 7.1 = 23.43 \text{ cm}^2$$ $$4.4 \times 7.1 = 31.24 \text{ cm}^2$$ $$5.5 \times 7.1 = 39.05 \text{ cm}^2$$ 5. **Calculate total surface area:** Total area = 2 × area of triangle + sum of areas of rectangles $$= 2 \times 7.26 + 23.43 + 31.24 + 39.05 = 14.52 + 93.72 = 108.24 \text{ cm}^2$$ 6. **Final answer:** The total surface area of the triangular prism is approximately **108.24 cm²**.