To find the 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 lateral surface area).
- Area of the triangular base:
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can use the width \( W \) as the base and the height \( H \) (the height of the triangle from base to the apex) for the triangular base:
\[ A = \frac{1}{2} \times 30.5 , \text{cm} \times 5.1 , \text{cm} \]
Calculating this:
\[ A = \frac{1}{2} \times 30.5 \times 5.1 = 77.775 , \text{cm}^2 \]
Since there are two triangular bases:
\[ \text{Total area of triangular bases} = 2 \times 77.775 = 155.55 , \text{cm}^2 \]
- Area of the rectangular sides:
Next, we compute the areas of the three rectangular sides. The three rectangles have the following dimensions:
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Rectangle 1 (base × length): \[ \text{Area}_1 = \text{width} \times \text{length} = 30.5 , \text{cm} \times 27.9 , \text{cm} = 852.45 , \text{cm}^2 \]
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Rectangle 2 (slant height × length): \[ \text{Area}_2 = \text{slant height} \times \text{length} = 28.4 , \text{cm} \times 27.9 , \text{cm} = 793.56 , \text{cm}^2 \]
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Rectangle 3 (height × length): \[ \text{Area}_3 = \text{height} \times \text{length} = 5.1 , \text{cm} \times 27.9 , \text{cm} = 142.29 , \text{cm}^2 \]
Now we add the areas of the three rectangular sides together:
\[ \text{Total area of rectangular sides} = 852.45 + 793.56 + 142.29 = 1788.30 , \text{cm}^2 \]
- Total surface area:
To find the total surface area \( S.A. \), we add the area of the triangular bases and the area of the rectangular sides:
\[ S.A. = \text{Total area of triangular bases} + \text{Total area of rectangular sides} \] \[ S.A. = 155.55 , \text{cm}^2 + 1788.30 , \text{cm}^2 = 1943.85 , \text{cm}^2 \]
Finally, rounding this to the nearest whole number, the total surface area is:
\[ \boxed{1944} , \text{cm}^2 \]
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