To find the surface area of the prism with a triangular base, we need to calculate two parts: the lateral surface area and the area of the two triangular bases.
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Area of the Triangular Base: The area \( A \) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] For the isosceles triangle with a base length of 12 inches and a height of 8 inches: \[ A = \frac{1}{2} \times 12 \times 8 = \frac{1}{2} \times 96 = 48 , \text{in}^2 \]
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Total Area of the Two Triangular Bases: Since the prism has two identical triangular bases, the area of both bases combined is: \[ \text{Area of both bases} = 2 \times A = 2 \times 48 = 96 , \text{in}^2 \]
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Lateral Surface Area: The lateral surface area of a prism is calculated by summing the areas of the rectangular sides. To find this, we determine the perimeter of the triangular base and multiply it by the height of the prism.
First, we calculate the lengths of the two equal sides of the isosceles triangle. Using the Pythagorean theorem:
- The base is 12 inches, so each equal side has a half-base of 6 inches.
- The height of 8 inches is the height of the triangle from the base to the apex. \[ \text{length of each equal side} = \sqrt{(6^2 + 8^2)} = \sqrt{36 + 64} = \sqrt{100} = 10 , \text{in} \]
Now we can find the perimeter \( P \) of the triangular base: \[ P = \text{base} + 2 \times \text{equal side} = 12 + 2 \times 10 = 12 + 20 = 32 , \text{in} \]
Thus, the lateral surface area \( L \) of the prism is: \[ L = P \times \text{height of the prism} = 32 \times 14 = 448 , \text{in}^2 \]
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Total Surface Area: Finally, the total surface area \( S \) of the prism is the sum of the lateral surface area and the areas of the two triangular bases: \[ S = L + \text{Area of both bases} = 448 + 96 = 544 , \text{in}^2 \]
Thus, the surface area of the prism is \( \boxed{544} \) in².
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