To find the surface area of the pyramid with a triangular base, we need to calculate both the area of the triangular base and the lateral surface area.
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Calculate the area of the triangular base: The triangular base has two equal sides, and we are given the height of the triangle.
The formula for the area \( A \) of a triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, we need to calculate the base. Since the triangle is isosceles with the two equal sides of 13 inches and the height from the apex down to the base being 12 inches, we can find the base using the Pythagorean theorem.
First, find the length of half the base (\( b/2 \)): \[ \text{Let } b/2 = x. \] Then by applying the Pythagorean theorem in one of the resulting right triangles: \[ 13^2 = 12^2 + x^2 \] \[ 169 = 144 + x^2 \] \[ x^2 = 25 \Rightarrow x = 5 \] Thus, the full base \( b \) is: \[ b = 2x = 2 \times 5 = 10 \text{ in.} \]
Now we can find the area of the base triangle: \[ A = \frac{1}{2} \times 10 \times 12 = 60 \text{ in}^2. \]
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Calculate the lateral surface area: The pyramid has three lateral triangular faces. Each triangular face has a base equal to the side of the triangular base and a height equal to the slant height of the pyramid.
The three sides of the triangular base are: \( 10 \) in. (the base side) and the two equal sides of \( 13 \) in. Let's consider the lateral surface area for the triangular face with the base length of \( 10 \) in. first: \[ A_{\text{lateral (base side)}} = \frac{1}{2} \times 10 \times 7 = 35 \text{ in}^2. \]
For the two lateral faces corresponding to the equal sides of \( 13 \) in.: \[ A_{\text{lateral (equal sides)}} = 2 \times \left(\frac{1}{2} \times 13 \times 7\right) = 2 \times \frac{91}{2} = 91 \text{ in}^2. \]
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Total lateral surface area: \[ A_{\text{lateral total}} = A_{\text{lateral (base side)}} + A_{\text{lateral (equal sides)}} = 35 + 91 = 126 \text{ in}^2. \]
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Total surface area of the pyramid: To find the total surface area \( A_{\text{total}} \), we add the area of the base to the lateral surface area: \[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral total}} = 60 + 126 = 186 \text{ in}^2. \]
Thus, the surface area of the pyramid is:
\[ \boxed{186 \text{ in}^2}. \]