Question

Find the slant height of the square pyramid. Round to the nearest whole inch, if necessary. Show your work.
The width is 18 in and the height is 12 in

Answer 1

To find the slant height of a square pyramid, we need to calculate the distance from the apex (top point) of the pyramid to the midpoint of one of its base edges.

1. **Identify the base dimensions and height**:
- The base is a square with a width of \(18\) inches.
- The height of the pyramid (the distance from the base to the apex) is \(12\) inches.

2. **Calculate the length of half the base (base radius)**:
\[
\text{Half of the width} = \frac{\text{Width}}{2} = \frac{18}{2} = 9 \text{ inches}
\]
This is the distance from the center of the base to the midpoint of one of the edges.

3. **Use the Pythagorean theorem to find the slant height**:
Let \(l\) be the slant height, which we want to find. According to the Pythagorean theorem:
\[
l^2 = \text{height}^2 + \left( \text{half of the base width} \right)^2
\]
Substituting in the values:
\[
l^2 = 12^2 + 9^2
\]
Calculate \(12^2\) and \(9^2\):
\[
12^2 = 144, \quad 9^2 = 81
\]
Therefore:
\[
l^2 = 144 + 81 = 225
\]

4. **Calculate the slant height \(l\)**:
\[
l = \sqrt{225} = 15 \text{ inches}
\]

Thus, the slant height of the square pyramid is \(\boxed{15}\) inches.