Short Answer 
The total surface area of the triangular pyramid is 238.5 square inches, calculated by adding the base area of 43.5 square inches and the lateral area of 195 square inches.
Step 1: Calculate the Base Area
The first step in finding the surface area of a triangular pyramid is to calculate the area of the base triangle. This is done using the formula:
- A = 1/2(a √ó b)
In this case, where a is the base length (10 units) and b is the height (8.7 units), you find:
- A = 1/2(10 √ó 8.7) = 43.5 square inches
Step 2: Calculate the Lateral Area
The next step is to compute the lateral area of the pyramid, which consists of three identical triangular faces. Each lateral triangular face can be calculated using:
- Lateral Area = 3(1/2 √ó b √ó s)
Where b is the base length (10 units) and s is the slant height (13 units), giving:
- Lateral Area = 3(1/2 √ó 10 √ó 13) = 3(65) = 195 square inches
Step 3: Add the Areas Together
The final step is to add the base area and the lateral area together to find the total surface area of the triangular pyramid:
- Total Surface Area = Base Area + Lateral Area
- Total Surface Area = 43.5 + 195
This results in a total surface area of:
- Total Surface Area = 238.5 square inches