To find the surface area of a pyramid with a triangular base, we need to calculate the area of the triangular base and the area of the three triangular faces.
Step 1: Calculate the area of the triangular base
The base of the pyramid is a triangle with two equal sides (an isosceles triangle) and the height from the apex of the triangle to the base side. The formula for the area \( A \) of a triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our isosceles triangle, we need to find the base. In this case, the height divides the base into two equal segments. Let the base be \( b \).
Using the Pythagorean theorem on one of the right triangles formed by the height (12 in) and half the base:
\[ (13)^2 = (b/2)^2 + (12)^2 \]
Calculating this:
\[ 169 = (b/2)^2 + 144 \] \[ (b/2)^2 = 169 - 144 \] \[ (b/2)^2 = 25 \] \[ b/2 = 5 \quad \Rightarrow \quad b = 10 \]
Now we can calculate the area of the triangular base:
\[ A = \frac{1}{2} \times 10 \times 12 = \frac{120}{2} = 60 \text{ in}^2 \]
Step 2: Calculate the area of the triangular faces
The pyramid has three triangular faces. The area of each triangular face can be calculated using the formula:
\[ A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the two triangular faces with the equal sides:
Each of these faces has a base equal to one side of the triangular base (10 in) and a slant height of 7 in.
Calculating the area of one of those faces:
\[ A_\text{face1} = \frac{1}{2} \times 10 \times 7 = \frac{70}{2} = 35 \text{ in}^2 \]
Since there are two such triangular faces:
\[ \text{Total area of two faces} = 2 \times 35 = 70 \text{ in}^2 \]
For the triangular face opposite the base:
This face has a base equal to one side of the triangular base (10 in) and a slant height of 7 in.
Since we already calculated this section, the area of the last triangular face would also be:
\[ A_\text{face2} = \frac{1}{2} \times 10 \times 7 = 35 \text{ in}^2 \]
Step 3: Calculate total surface area
Now we find the total surface area by adding the area of the base and the areas of the three triangular faces:
\[ \text{Total Surface Area} = A_\text{base} + A_\text{face1} + A_\text{face2} \]
With two faces having the same dimensions, we compute:
\[ \text{Total Surface Area} = 60 \ + \ (35 + 35 + 35) = 60 + 105 = 165 \text{ in}^2 \]
Thus, the total surface area of the pyramid is:
\[ \boxed{165} \text{ in}^2 \]