To calculate the total surface area of a triangular prism, we need to find the area of the triangular faces and the areas of the three rectangular faces.
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Area of the Triangular Face: If we denote the base of the triangle (b) and the height (h) of the triangle, the area (A) of one triangular face can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
We need the base and height dimensions to compute this, but let's assume hypothetical values based on the problem statement (you would replace these with the corresponding values from the figure):
Let’s assume:
- Base \(b = 28.4 , cm\)
- Height \(h = 5.1 , cm\)
Thus, \[ A = \frac{1}{2} \times 28.4 , cm \times 5.1 , cm = \frac{1}{2} \times 145.284 , cm^2 = 72.642 , cm^2 \]
Area of each triangular face: 72.642 cm² (since there are two triangular faces, it would be \(2 \times 72.642 , cm^2\)).
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Areas of the Rectangular Faces: Assume the lengths of the rectangular faces are based on the dimensions given.
Let’s say:
- The smallest rectangular face is opposite the smallest base of the triangle, with height equal to the prism height (assumed to be \(5.1 , cm\) and base length \(28.4, cm\)): \[ \text{Area (smallest rectangular face)} = \text{base} \times \text{height} = 27.9 ,cm \times \text{Prism Height} \]
Let’s assume the height of the prism = H. \[ \text{Area (smallest)} = 27.9,cm \times H \]
For the middle rectangular face (base \(= 28.4, cm\)) and largest rectangular face (base \(= 30.5, cm\)):
- Middle rectangular face (19.7 cm): \[ \text{Area (largest rectangular face)} = \text{base} \times \text{height} = 30.5 , cm \times H \]
Total Surface Area Calculation:
Putting these expressions together: \[ \text{Total Surface Area} = 2 \times \text{Area of Triangular Face} + \text{Area of Smallest Rectangular Face} + \text{Area of Middle Rectangular Face} + \text{Area of Largest Rectangular Face} \]
If we manage to define all height values (let's say H = 30.5 cm), \[ \text{Total Surface Area} = 2 \times 72.642 , cm^2 + (27.9, cm * H ) + (29.5, cm * H ) + (30.5, cm * H ) \]
You would input circumstantial numbers for this height ensuring a scalable increase through orientations and indicate final calculations based on realistic scenarios.
Complete this calculation by replacing the \(H\) with the actual height of the prism, and sum everything together.
Feel free to provide additional details from the figure containing data, and I can help with precise values too.