To find the surface area of a triangular prism, we need to calculate the area of the triangular base and the areas of the three rectangular faces.
Step 1: Find the area of the triangular base
The base of the prism is a triangle, and we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base \( W \) is given as 4 cm, and the height \( H \) is given as 3 cm.
\[ \text{Area}_{\text{triangular base}} = \frac{1}{2} \times 4 , \text{cm} \times 3 , \text{cm} = 6 , \text{cm}^2 \]
Step 2: Calculate the area of the rectangular faces
The triangular prism has three rectangular faces and two triangular bases. The dimensions of the rectangular faces can be calculated as follows:
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Face 1 (base rectangular face): \[ \text{Area}_{\text{rectangular face}} = L \times W = 11 , \text{cm} \times 4 , \text{cm} = 44 , \text{cm}^2 \]
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Face 2 (height rectangular face): \[ \text{Area}_{\text{rectangular face}} = H \times L = 3 , \text{cm} \times 11 , \text{cm} = 33 , \text{cm}^2 \]
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Face 3 (slant height rectangular face): \[ \text{Area}_{\text{rectangular face}} = \text{Slant Height} \times \text{Base} = 5 , \text{cm} \times 4 , \text{cm} = 20 , \text{cm}^2 \]
Step 3: Sum the areas
Now we can sum the areas of the two triangular bases and the three rectangular faces to find the total surface area of the triangular prism.
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Total area of triangular bases: \[ 2 \times \text{Area}_{\text{triangular base}} = 2 \times 6 , \text{cm}^2 = 12 , \text{cm}^2 \]
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Total area of rectangular faces: \[ \text{Area}{\text{rectangular face 1}} + \text{Area}{\text{rectangular face 2}} + \text{Area}_{\text{rectangular face 3}} = 44 , \text{cm}^2 + 33 , \text{cm}^2 + 20 , \text{cm}^2 = 97 , \text{cm}^2 \]
Step 4: Final surface area
Finally, combine the areas of the triangular bases and the rectangular faces:
\[ \text{Total Surface Area} = 12 , \text{cm}^2 + 97 , \text{cm}^2 = 109 , \text{cm}^2 \]
Thus, the surface area of the triangular prism is 109 square centimeters.