To find the surface area of a triangular prism, we need to determine the areas of the triangular bases and the three rectangular sides.
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Calculate the area of the triangular base: The base of the triangle is 4 cm, and the height (rise) is 3 cm. The area \( A \) of the triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \] \[ A = \frac{1}{2} \times 4 , \text{cm} \times 3 , \text{cm} = \frac{12}{2} , \text{cm}^2 = 6 , \text{cm}^2 \]
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Calculate the lateral surface area: The three rectangular sides of the prism correspond to the lengths:
- One rectangular side with dimensions of the base and the length of the prism: \[ \text{Area}_1 = \text{base} \times \text{length} = 4 , \text{cm} \times 11 , \text{cm} = 44 , \text{cm}^2 \]
- The second rectangular side with dimensions of the rise and the length: \[ \text{Area}_2 = \text{rise} \times \text{length} = 3 , \text{cm} \times 11 , \text{cm} = 33 , \text{cm}^2 \]
- The third rectangular side is the sloped side, which has a length of 5 cm: \[ \text{Area}_3 = \text{slope} \times \text{length} = 5 , \text{cm} \times 11 , \text{cm} = 55 , \text{cm}^2 \]
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Sum the areas: The total lateral surface area is: \[ \text{Lateral Surface Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 \] \[ = 44 , \text{cm}^2 + 33 , \text{cm}^2 + 55 , \text{cm}^2 = 132 , \text{cm}^2 \]
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Add the areas of the two triangular bases: Since there are two identical triangular bases, the area contributed by the bases is: \[ \text{Area of two triangular bases} = 2 \times A = 2 \times 6 , \text{cm}^2 = 12 , \text{cm}^2 \]
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Total surface area: Finally, the total surface area \( S \) of the prism is the sum of the lateral surface area and the area of the triangular bases: \[ S = \text{Lateral Surface Area} + \text{Area of two triangular bases} \] \[ S = 132 , \text{cm}^2 + 12 , \text{cm}^2 = 144 , \text{cm}^2 \]
Thus, the total surface area of the triangular prism is 144 cm².