Question

calculate the surface area of the square prism that has a base length of 12 in a slant height of 18 show your work

Answer 1

To calculate the surface area of a square prism (also known as a square pyramid if it has a point at the top), we need to clarify the shapes involved. I will assume you are referring to a square pyramid with a square base.

1. **Identify the dimensions**:
- Base length of the square (b): 12 inches
- Slant height (l): 18 inches

2. **Calculate the area of the base**:
\[
\text{Area of the base} = b^2 = 12^2 = 144 \text{ square inches}
\]

3. **Calculate the area of the four triangular faces**:
Each triangular face has a base equal to the side length of the square (12 inches) and a height equal to the slant height (18 inches).

\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 18 = \frac{1}{2} \times 216 = 108 \text{ square inches}
\]

Since there are four triangular faces:
\[
\text{Total area of the triangular faces} = 4 \times 108 = 432 \text{ square inches}
\]

4. **Calculate the total surface area**:
The total surface area (SA) of the pyramid combines the area of the base and the area of the triangular faces:
\[
\text{Total Surface Area} = \text{Area of the base} + \text{Area of the triangular faces}
\]
\[
\text{Total Surface Area} = 144 + 432 = 576 \text{ square inches}
\]

Therefore, the surface area of the square pyramid is **576 square inches**.