To find the total surface area of a triangular prism, we need to calculate the area of the two triangular bases and the area of the three rectangular sides. The formulas used are:
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Area of the triangular base: The area \( A \) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base is given as 6 cm and the height (h) is 4 cm: \[ A = \frac{1}{2} \times 6 \times 4 = \frac{24}{2} = 12 , \text{cm}^2 \] Since there are two triangular bases, the total area of the triangular bases is: \[ 2 \times 12 = 24 , \text{cm}^2 \]
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Area of the rectangular sides: The prism has three rectangular faces. The areas of these rectangles can be calculated as follows:
- Rectangle 1 (base * side): \[ \text{Area}_1 = \text{base length} \times \text{slant height} = 6 \times 5 = 30 , \text{cm}^2 \]
- Rectangle 2 (side length * height): \[ \text{Area}_2 = \text{side length} \times \text{slant height} = 12 \times 5 = 60 , \text{cm}^2 \]
- Rectangle 3 (height * slant height): \[ \text{Area}_3 = \text{height} \times \text{slant height} = 4 \times 5 = 20 , \text{cm}^2 \]
Now, summing up the area of the rectangular sides: \[ \text{Total rectangular areas} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 30 + 60 + 20 = 110 , \text{cm}^2 \]
- Total Surface Area: Finally, we calculate the total surface area by adding the area of the triangular bases and the area of the rectangular faces: \[ \text{Total Surface Area} = \text{Total triangular areas} + \text{Total rectangular areas} = 24 + 110 = 134 , \text{cm}^2 \]
From your provided options, it seems there may have been confusion, as none match the calculated value. Ensure that the parameters provided are consistent with a typical triangular prism calculation. Please double-check your dimensions or consult your instructor for possible clarification.
Therefore, the total surface area of the triangular prism should be \( 134 , \text{cm}^2 \).