To find the surface area of a triangular prism, we need to know the area of the triangular base and the area of the three rectangular faces.
Let's break it down step by step.
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Calculate the area of the triangular base: The base of the triangle can be considered as \( W \) and \( H \), with \( L \) being the height from the base to the vertex opposite the base.
The triangle has a base \( W = 4 , \text{cm} \) and height \( H = 3 , \text{cm} \).
The area \( A \) of the triangular base is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times W \times H = \frac{1}{2} \times 4 \times 3 = 6 , \text{cm}^2 \]
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Calculate the areas of the rectangular faces: A triangular prism has three rectangular faces. The dimensions of these rectangles are:
- Face 1 (from the triangle's base): \[ \text{Area} = W \times L = 4 \times 11 = 44 , \text{cm}^2 \]
- Face 2 (from the height of the triangle): \[ \text{Area} = H \times L = 3 \times 11 = 33 , \text{cm}^2 \]
- Face 3 (from the slant height of the triangle): \[ \text{Area} = \text{slant } H \times L = 5 \times 11 = 55 , \text{cm}^2 \]
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Total surface area: The total surface area \( SA \) of the prism is the sum of the areas of the two triangular bases (since there are two identical bases), and the areas of the three rectangular faces: \[ SA = 2 \times A + \text{Area of the 3 faces} \] \[ SA = 2 \times 6 + 44 + 33 + 55 \] \[ SA = 12 + 44 + 33 + 55 = 144 , \text{cm}^2 \]
Thus, the surface area of the triangular prism is 144 square centimeters.