To find the surface area of a triangular prism, we need to know the dimensions of both the triangular base and the height of the prism.
From your message, it seems like we have:
- The lengths of the sides of the triangular base: \( a = 6 , \text{cm} \), \( b = 8 , \text{cm} \), and \( c = 10 , \text{cm} \).
- The height of the prism: \( h = 10 , \text{cm} \).
Step 1: Calculate the area of the triangular base
To find the area of the triangle, we can use Heron's formula:
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Calculate the semi-perimeter \( s \): \[ s = \frac{a + b + c}{2} = \frac{6 + 8 + 10}{2} = 12 , \text{cm} \]
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Calculate the area \( A \) using Heron's formula: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] \[ A = \sqrt{12(12 - 6)(12 - 8)(12 - 10)} = \sqrt{12 \times 6 \times 4 \times 2} \] \[ = \sqrt{576} = 24 , \text{cm}^2 \]
Step 2: Calculate the surface area of the prism
The surface area \( SA \) of the triangular prism can be calculated using the formula: \[ SA = 2A + P \cdot h \] Where \( P \) is the perimeter of the triangular base.
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Calculate the perimeter \( P \): \[ P = a + b + c = 6 + 8 + 10 = 24 , \text{cm} \]
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Now substitute the values into the surface area formula: \[ SA = 2 \times A + P \cdot h = 2 \times 24 + 24 \times 10 \] \[ = 48 + 240 = 288 , \text{cm}^2 \]
Final Answer
The surface area of the triangular prism is \( 288 , \text{cm}^2 \).