To find the surface area of a triangular prism, we need to calculate the area of the triangular bases and the area of the three rectangular faces.
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Area of the triangular base: The base is an equilateral triangle with side length \( s = 6 \) inches. The area \( A \) of an equilateral triangle can be calculated using the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \] Plugging in the side length: \[ A = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3} \text{ square inches} \]
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Area of the two triangular bases: Since there are two triangular bases, the total area of the triangular bases is: \[ 2A = 2 \times 9\sqrt{3} = 18\sqrt{3} \text{ square inches} \]
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Area of the rectangular faces: The prism has three rectangular sides, and we have to calculate their areas:
- The height of the prism is \( h = 10 \) inches.
- The lengths of the sides of the triangular base are all \( 6 \) inches.
The areas of the three rectangular faces are calculated as follows:
- First rectangle (base side 6 inches): \[ \text{Area}_1 = \text{Base side} \times \text{Height} = 6 \times 10 = 60 \text{ square inches} \]
- Second rectangle (base side 6 inches): \[ \text{Area}_2 = 6 \times 10 = 60 \text{ square inches} \]
- Third rectangle (base side 6 inches): \[ \text{Area}_3 = 6 \times 10 = 60 \text{ square inches} \]
Thus, the total area of the rectangular faces is: \[ \text{Total Rectangular Area} = 60 + 60 + 60 = 180 \text{ square inches} \]
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Total surface area of the prism: Finally, the total surface area \( SA \) of the triangular prism is the sum of the area of the triangular bases and the area of the rectangular faces: \[ SA = \text{Area of the triangular bases} + \text{Area of the rectangular faces} \] \[ SA = 18\sqrt{3} + 180 \]
So, the surface area of the triangular prism is: \[ SA = 18\sqrt{3} + 180 \text{ square inches} \]
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