To calculate the total surface area of a triangular prism, we first need to find the area of the triangular faces and the three rectangular faces. Here, we have a triangular prism described by several dimensions:
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Triangular Faces
- Let's denote the lengths of the sides of the triangular base as \(a = 30.5 , \text{cm}\), \(b = 28.4 , \text{cm}\), and \(c = 27.9 , \text{cm}\).
- We need to calculate the area of the triangular face using Heron's formula.
- The semi-perimeter \(s\) of the triangle is calculated as follows: \[ s = \frac{a + b + c}{2} = \frac{30.5 + 28.4 + 27.9}{2} = \frac{86.8}{2} = 43.4 , \text{cm} \]
- The area \(A\) of the triangle can then be calculated using the formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] \[ = \sqrt{43.4(43.4 - 30.5)(43.4 - 28.4)(43.4 - 27.9)} \] \[ = \sqrt{43.4 \times 12.9 \times 15 \times 15.5} \]
- First, we find the values: \[ 43.4 - 30.5 = 12.9, \quad 43.4 - 28.4 = 15, \quad 43.4 - 27.9 = 15.5 \]
- Now, substituting those values in: \[ A = \sqrt{43.4 \times 12.9 \times 15 \times 15.5} \]
We can calculate the product step by step:
- \( 43.4 \times 12.9 = 560.46 \)
- \( 15 \times 15.5 = 232.5 \)
- Now multiply: \[ 560.46 \times 232.5 = 130,889.37 \]
- Now we compute the square root: \[ A \approx \sqrt{130889.37} \approx 362.5 , \text{cm}^2 \]
Therefore, the area of each triangular face is approximately \(\text{362.5 cm}^2\).
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Rectangular Faces
- Let the height of the prism be \(h = 5.1 , \text{cm}\).
- The areas of the three rectangular faces can be calculated by multiplying the height by the lengths of the respective sides of the triangle:
- For the smallest rectangular face (side \(a\)): \[ \text{Area of smallest rectangular face} = a \cdot h = 30.5 , \text{cm} \times 5.1 , \text{cm} = 155.55 , \text{cm}^2 \]
- For the middle rectangular face (side \(b\)): \[ \text{Area of middle rectangular face} = b \cdot h = 28.4 , \text{cm} \times 5.1 , \text{cm} = 145.84 , \text{cm}^2 \]
- For the largest rectangular face (side \(c\)): \[ \text{Area of largest rectangular face} = c \cdot h = 27.9 , \text{cm} \times 5.1 , \text{cm} = 142.29 , \text{cm}^2 \]
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Total Surface Area
- Now we can calculate the total surface area of the prism: \[ \text{Total Surface Area} = 2 \cdot \text{Area of Triangular Face} + \text{Area of Smallest Rectangular Face} + \text{Area of Middle Rectangular Face} + \text{Area of Largest Rectangular Face} \] \[ \text{Total Surface Area} = 2 \cdot 362.5 + 155.55 + 145.84 + 142.29 \]
- Calculating this gives: \[ = 725 + 155.55 + 145.84 + 142.29 = 1168.68 , \text{cm}^2 \]
So, here are the final values:
- Area of each triangular face = \( \approx 362.5 , \text{cm}^2 \)
- Area of the smallest rectangular face = \( 155.55 , \text{cm}^2 \)
- Area of the middle rectangular face = \( 145.84 , \text{cm}^2 \)
- Area of the largest rectangular face = \( 142.29 , \text{cm}^2 \)
- TOTAL Surface Area of the Triangular Prism = \( 1168.68 , \text{cm}^2 \)