Question

Determine the surface area of a pyramid with a triangular base. The triangular base has two equal sides of 13 inches. The height of the base side is 12 inches and the slant height is 7 in

Answer 1

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

1. **Calculate the area of the triangular base**:
The base of the pyramid is an isosceles triangle with two equal sides of 13 inches and a height of 12 inches.
The area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we need to find the base length. To do that, we can split the triangle into two right triangles.

Each half of the base can be found using:
\[
b = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5
\]
Therefore, the entire base length is \( 2b = 2 \times 5 = 10 \) inches.

Now we can calculate the area of the triangular base:
\[
A_{\text{base}} = \frac{1}{2} \times 10 \text{ in} \times 12 \text{ in} = 60 \text{ in}^2
\]

2. **Calculate the area of the lateral triangular faces**:
The pyramid has 3 triangular lateral faces. The area of each face can be calculated using:
\[
A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{slant height}
\]

For each face, since the base is 10 inches (the base length of the triangular base), the slant height is given as 7 inches.
The two lateral faces that are opposite each other will use the base of 5 inches (half the base of the triangular base), and the final face will also use 5 inches.

- The area for the two equivalent faces:
\[
A_{\text{lateral}} = 2 \times \left(\frac{1}{2} \times 5 \text{ in} \times 7 \text{ in}\right) = 2 \times \left(\frac{35}{2}\right) = 35 \text{ in}^2
\]

- The area for the base face:
\[
A_{\text{base face}} = \frac{1}{2} \times 10 \text{ in} \times 7 \text{ in} = 35 \text{ in}^2
\]

3. **Adding up the total area**:
The total surface area of the pyramid is:
\[
A_{\text{total}} = A_{\text{base}} + 2 \times A_{\text{face}} + A_{\text{base face}}
\]
\[
A_{\text{total}} = 60 \text{ in}^2 + 35 \text{ in}^2 + 35 \text{ in}^2 = 130 \text{ in}^2
\]

So, the total surface area of the pyramid is \( 130 \) square inches.