Question

Look at the square pyramid below.



40 mi40\ \text{mi}40 mi 𝑙ll

9 mi9\ \text{mi}9 mi

How many miles long is the slant height of the square pyramid? Click Here for Help Video.

Answer 1

To find the slant height of the square pyramid, we can use the Pythagorean theorem.

1. **Identify the values**:
- The base of the pyramid is a square with each side measuring 40 miles.
- The height of the pyramid (vertical height from the base to the apex) is 9 miles.

2. **Find the length of half the base**:
- Since the base is square, half of one side of the base is \( \frac{40}{2} = 20 \) miles.

3. **Use the Pythagorean theorem**:
The slant height \( l \) forms a right triangle with half the base and the height of the pyramid. Therefore, we can use the equation:

\[
l^2 = \text{height}^2 + \left(\frac{\text{base}}{2}\right)^2
\]

Plugging in the values:

\[
l^2 = 9^2 + 20^2
\]

\[
l^2 = 81 + 400
\]

\[
l^2 = 481
\]

4. **Calculate the slant height**:
Take the square root:

\[
l = \sqrt{481}
\]

Approximating this gives:

\[
l \approx 21.93 \text{ miles}
\]

Thus, the slant height of the square pyramid is approximately **21.93 miles**.