To find the surface area of a triangular prism, we need to know the area of the triangular base and the lateral surface area.
Given the dimensions:
- The triangle has sides 8 cm, 6 cm, and 9 cm.
- The height (length) of the prism is 10 cm.
Step 1: Calculate the Area of the Triangular Base
First, we can use Heron's formula to find the area of the triangle.
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Find the semi-perimeter \( s \): \[ s = \frac{a + b + c}{2} = \frac{8 + 6 + 9}{2} = \frac{23}{2} = 11.5 , \text{cm} \]
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Use Heron's formula: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] Substituting the values: \[ A = \sqrt{11.5(11.5 - 8)(11.5 - 6)(11.5 - 9)} \] \[ A = \sqrt{11.5 \times 3.5 \times 5.5 \times 2.5} \] \[ A = \sqrt{11.5 \times 3.5 \times 5.5 \times 2.5} \approx \sqrt{11.5 \times 3.5 \times 5.5 \times 2.5} \approx \sqrt{500.6875} \approx 22.36 , \text{cm}^2 \]
Step 2: Calculate the Lateral Surface Area
The lateral surface area of the prism is the sum of the areas of the three rectangular sides:
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The three sides are given by:
- base side \( 8 , \text{cm} \)
- side \( 6 , \text{cm} \)
- side \( 9 , \text{cm} \)
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The lateral surface areas are calculated as follows:
- Area of the rectangle corresponding to side \( 8 \): \( 8 \times 10 = 80 , \text{cm}^2 \)
- Area of the rectangle corresponding to side \( 6 \): \( 6 \times 10 = 60 , \text{cm}^2 \)
- Area of the rectangle corresponding to side \( 9 \): \( 9 \times 10 = 90 , \text{cm}^2 \)
Adding these, the total lateral surface area is: \[ 80 + 60 + 90 = 230 , \text{cm}^2 \]
Step 3: Calculate Total Surface Area
Now, we can find the total surface area by adding the area of the two triangular bases (which is \( 2A \)) to the lateral surface area.
\[ \text{Total Surface Area} = 2A + \text{Lateral Surface Area} \] \[ = 2 \times 22.36 + 230 \approx 44.72 + 230 = 274.72 , \text{cm}^2 \]
Conclusion
The total surface area of the triangular prism is approximately \( 274.72 , \text{cm}^2 \).