To calculate the surface area of a pyramid with a triangular base, we need to find the area of the triangular base and the area of the three triangular lateral faces.
Step 1: Find the area of the triangular base.
The base of the pyramid is an isosceles triangle, given that the two equal sides are 13 in each and the height from the apex of the triangle down to the base is 12 in.
The area \( A \) of the triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] To find the base of the triangle, we can use the Pythagorean theorem in one of the right triangles formed by dropping a height from the apex to the base. Since the triangle is isosceles, the height divides the base into two equal segments.
Let \( x \) be half of the base. Using the Pythagorean theorem: \[ 13^2 = 12^2 + x^2 \] \[ 169 = 144 + x^2 \] \[ x^2 = 169 - 144 = 25 \] \[ x = \sqrt{25} = 5 \] Thus, the entire base of the triangle is \( 2x = 2(5) = 10 \) in.
Now we can calculate the area of the triangle: \[ A = \frac{1}{2} \times 10 \times 12 = 60 \text{ in}^2 \]
Step 2: Calculate the area of the lateral faces.
There are three lateral triangular faces. The area of each triangular face can be found using the formula: \[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the two triangular faces that correspond to the equal sides of length 13 in and the slant height of 7 in:
- The base is 10 in (the base of the triangular base).
- The height (slant height of the pyramid) is 7 in.
So, the area of each of these triangular faces will be: \[ A_1 = \frac{1}{2} \times 10 \times 7 = 35 \text{ in}^2 \]
Since there are two of these triangular faces: \[ A_{\text{two faces}} = 2 \times 35 = 70 \text{ in}^2 \]
For the third triangular face, which corresponds to the apex connecting back to the base's two ends (the base being the 10 in side), the area is as follows. The base here is 10 in and the height (slant height) is 7 in: \[ A_2 = \frac{1}{2} \times 10 \times 7 = 35 \text{ in}^2 \]
Step 3: Calculate the total surface area.
Sum the areas:
Total surface area \( SA \) of the pyramid: \[ SA = \text{Area of base} + \text{Area of lateral faces} \] \[ SA = 60 + 70 + 35 \] \[ SA = 60 + 105 = 165 \text{ in}^2 \]
Thus, the total surface area of the pyramid is: \[ \boxed{165 \text{ in}^2} \]