To calculate the surface area (S.A.) of a triangular prism, we need to find the area of the two triangular bases and the three rectangular sides.
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Find the area of the triangular face (base area): The triangular face is a right triangle, where:
- Base (b) = 27.9 cm
- Height (h) = 5.1 cm
The area of a triangle is given by the formula: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 27.9 \times 5.1 \approx \frac{1}{2} \times 142.29 \approx 71.145 , \text{cm}^2 \]
Since the prism has two identical triangular bases, the total area for the triangular bases is: \[ \text{Area}_{\text{triangles}} = 2 \times 71.145 \approx 142.29 , \text{cm}^2 \]
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Find the area of the rectangular sides: The prism has three rectangular sides which can be calculated as follows:
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Side 1 (base side): Dimensions = base × length = 27.9 cm × 30.5 cm \[ \text{Area}_{\text{base}} = 27.9 \times 30.5 \approx 850.95 , \text{cm}^2 \]
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Side 2 (height side): Dimensions = height × length = 5.1 cm × 30.5 cm \[ \text{Area}_{\text{height}} = 5.1 \times 30.5 \approx 155.55 , \text{cm}^2 \]
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Side 3 (hypotenuse side): Dimensions = hypotenuse × length = 28.4 cm × 30.5 cm \[ \text{Area}_{\text{hypotenuse}} = 28.4 \times 30.5 \approx 867.2 , \text{cm}^2 \]
The total area of the rectangular sides is: \[ \text{Area}_{\text{rectangles}} = 850.95 + 155.55 + 867.2 \approx 1873.7 , \text{cm}^2 \]
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Total surface area: Now, we add the area of the triangular bases and the area of the rectangular sides to get the total surface area of the prism: \[ \text{S.A.} = \text{Area}{\text{triangles}} + \text{Area}{\text{rectangles}} \] \[ \text{S.A.} = 142.29 + 1873.7 \approx 2016 , \text{cm}^2 \]
Therefore, the amount of wrapping paper needed to wrap the toy is approximately:
\[ \boxed{2016} , \text{cm}^2 \]