To solve this problem, we first need to clarify the components of the triangular prism based on the information provided.
Step 1: Determine the number of faces
A triangular prism consists of:
- 2 triangular faces (top and bottom)
- 3 rectangular faces (sides)
Thus, the total number of faces is: Total faces = 2 + 3 = 5 faces.
Step 2: Calculate the area of each triangular face
Given the side lengths of the triangle are 10 cm, 8 cm, 9 cm, and another side at 6 cm, it is assumed that the relevant triangle for the prism is specified with sides 10 cm, 8 cm, and 9 cm.
We can use Heron's formula to find the area of the triangle.
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Calculate the semi-perimeter \( s \): \[ s = \frac{10 + 8 + 9}{2} = \frac{27}{2} = 13.5 , \text{cm} \]
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Calculate the area \( A \) using Heron's formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \( a = 10 , \text{cm}, b = 8 , \text{cm}, c = 9 , \text{cm} \): \[ A = \sqrt{13.5(13.5 - 10)(13.5 - 8)(13.5 - 9)} \]
Calculate each term: \[ A = \sqrt{13.5 \times 3.5 \times 5.5 \times 4.5} \]
We approximate this by calculating the quantities: \[ 13.5 \times 3.5 = 47.25 \] \[ 5.5 \times 4.5 = 24.75 \] \[ A = \sqrt{47.25 \times 24.75} \approx \sqrt{1165.6875} \approx 34.14 , \text{cm}^2 \]
Area of each triangular face: Area = 34.14 square centimeters (this value may need adjustment based on exact values).
Step 3: Calculate the areas of the rectangular faces
Assuming the height or length of the prism (the perpendicular distance between the triangular faces) is not given, we generally denote this height as \( h \). We need to calculate the areas of each rectangular face:
- Smallest rectangular face (base 6 cm): \[ \text{Area} = \text{base} \times \text{height} = 6h , \text{cm}^2 \]
- Middle rectangular face (base 8 cm): \[ \text{Area} = 8h , \text{cm}^2 \]
- Largest rectangular face (base 10 cm): \[ \text{Area} = 10h , \text{cm}^2 \]
Step 4: Calculate the total surface area
The total surface area \( TSA \) of the triangular prism is given by: \[ TSA = 2 \times \text{Area of triangular faces} + \text{Area of rectangular faces} \] Substituting values: \[ TSA = 2 \times 34.14 + (6h + 8h + 10h) \] \[ TSA = 68.28 + 24h , \text{cm}^2 \]
Thus, to summarize your answers in terms of the required format:
- Number of faces: 5 faces
- Area of each triangular face: Approximately 34.14 square centimeters
- Area of the smallest rectangular face: \( 6h \) square centimeters
- Area of the middle rectangular face: \( 8h \) square centimeters
- Area of the largest rectangular face: \( 10h \) square centimeters
- Total surface area of the triangular prism: \( 68.28 + 24h \) square centimeters
Please adjust the value of \( h \) based on your specific prism dimensions to calculate exact values for rectangular face areas and the total surface area.