Question

Find the surface area of a triangular pyramid with base 10, a base height of 8.7, and a slant height of 12.(1 point)
Responses

103.5
103.5

232.2
232.2

216.6
216.6

223.5

Answer 1

To find the surface area of the triangular pyramid, we need to calculate the area of the triangular base and the area of the three triangular faces.

1. **Area of the base (a triangle)**:
The formula for the area of a triangle is:
\[
\text{Area}_{base} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is 10 and the height is 8.7.
\[
\text{Area}_{base} = \frac{1}{2} \times 10 \times 8.7 = 43.5
\]

2. **Area of the lateral faces**:
Each lateral face is a triangle with a base of the side of the triangular base (which is 10 for the given problem) and a height of the slant height (12). Since all sides are not provided, and considering a equilateral triangle for this case where each side is equal to the base of 10, we calculate the area of one triangular face first:
\[
\text{Area}_{face} = \frac{1}{2} \times \text{base} \times \text{slant height}
\]
Using a side of 10 and slant height of 12:
\[
\text{Area}_{face} = \frac{1}{2} \times 10 \times 12 = 60
\]
Since there are three triangular faces, multiply by 3:
\[
\text{Total Area of faces} = 3 \times 60 = 180
\]

3. **Total surface area**:
Now, add the area of the base to the total area of the faces:
\[
\text{Total Surface Area} = \text{Area}_{base} + \text{Total Area of faces}
\]
\[
\text{Total Surface Area} = 43.5 + 180 = 223.5
\]

Thus, the surface area of the triangular pyramid is **223.5**. So the correct response is:

223.5