To calculate the surface area of an equilateral triangular prism, we need to consider both the lateral surface area and the area of the two triangular bases.
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Base Area: The base of the prism is an equilateral triangle. If the side length of the triangle is \( s \), the area (\( A \)) of one equilateral triangle can be calculated using the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \]
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Lateral Surface Area: The lateral surface area (the area of the three rectangular sides) can be calculated by multiplying the perimeter of the triangular base by the height (\( h \)) of the prism. The perimeter \( P \) of an equilateral triangle with side length \( s \) is: \[ P = 3s \] Thus, the lateral surface area (\( LSA \)) is: \[ LSA = P \times h = 3s \times h = 3sh \]
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Total Surface Area: The total surface area (\( SA \)) of the prism can be found by adding the lateral surface area and the areas of the two bases: \[ SA = LSA + 2 \times A \] Putting it all together: \[ SA = 3sh + 2 \left( \frac{\sqrt{3}}{4} s^2 \right) \]
Simplifying the equation gives us: \[ SA = 3sh + \frac{\sqrt{3}}{2} s^2 \]
Thus, the final equation for the surface area of an equilateral triangular prism is: \[ SA = 3sh + \frac{\sqrt{3}}{2} s^2 \]