To determine the surface area of the pyramid, 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 is an isosceles triangle with two equal sides measuring 13 inches and a height of 12 inches. The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
We need the base of the triangle. We can find the base using the properties of the isosceles triangle. Since the height divides the base into two equal segments, we can use the Pythagorean theorem to find half the base.
Let \( b \) be the length of the base. The height creates two right triangles:
\[ \left( \frac{b}{2} \right)^2 + 12^2 = 13^2 \] \[ \left( \frac{b}{2} \right)^2 + 144 = 169 \] \[ \left( \frac{b}{2} \right)^2 = 169 - 144 \] \[ \left( \frac{b}{2} \right)^2 = 25 \] \[ \frac{b}{2} = 5 \quad \Rightarrow \quad b = 10 \]
Now we can calculate the area of the triangular base:
\[ A = \frac{1}{2} \times 10 \times 12 = 60 , \text{square inches} \]
Step 2: Calculate the area of the triangular faces.
The pyramid has three triangular faces. We need to find the area of each triangular face.
- Two faces with base = 13 in and slant height = 7 in:
The area of one triangle face is given by:
\[ A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{slant height} \]
For the two triangular faces that have bases of length 13 inches:
\[ A_{\text{face}} = \frac{1}{2} \times 13 \times 7 = \frac{91}{2} = 45.5 , \text{square inches} \]
Since there are two such faces, the total area contributed by these faces is:
\[ 2 \times 45.5 = 91 , \text{square inches} \]
- One face with base = 10 in and slant height = 7 in:
For the face that has a base of 10 inches:
\[ A_{\text{face}} = \frac{1}{2} \times 10 \times 7 = 35 , \text{square inches} \]
Step 3: Calculate the total surface area of the pyramid.
The total surface area \( S \) of the pyramid is the sum of the area of the base and the areas of the triangular faces:
\[ S = \text{Area of base} + \text{Area of triangular faces} \]
\[ S = 60 + 91 + 35 = 186 , \text{square inches} \]
Final Answer:
The surface area of the pyramid is
\[ \boxed{186} , \text{square inches}. \]