Question

Determine the surface area of a pyramid with a square base. The side of the square base is 3 in. and the slant height of the pyramid is 5 in.

Answer 1

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

1. **Area of the Base**:
The base is a square with a side length of 3 inches. The area \( A \) of the base is given by the formula:
\[
A_{\text{base}} = s^2
\]
where \( s \) is the side length of the square. Here, \( s = 3 \) in, so:
\[
A_{\text{base}} = 3^2 = 9 \text{ square inches}
\]

2. **Area of the Triangular Faces**:
The pyramid has 4 triangular faces. The area \( A \) of one triangular face is given by the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \cdot \text{base} \cdot \text{height}
\]
For our triangular faces, the base of each triangle is the same as the side length of the square base, which is 3 inches, and the height of each triangle is the slant height of the pyramid, which is 5 inches. Thus, the area of one triangular face is:
\[
A_{\text{triangle}} = \frac{1}{2} \cdot 3 \cdot 5 = \frac{15}{2} = 7.5 \text{ square inches}
\]

Since there are 4 triangular faces, the total area of the triangular faces \( A_{\text{triangles}} \) is:
\[
A_{\text{triangles}} = 4 \cdot A_{\text{triangle}} = 4 \cdot 7.5 = 30 \text{ square inches}
\]

3. **Total Surface Area**:
Now, we can add the area of the base and the area of the triangular faces to get the total surface area \( A_{\text{total}} \):
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangles}} = 9 + 30 = 39 \text{ square inches}
\]

Thus, the total surface area of the pyramid is:
\[
\boxed{39 \text{ square inches}}
\]