Question

Lateral face: 56
Base: 33
Length: ???
Square pyramid
How many miles long is the slant height of the square pyramid?
only as an integer

Answer 1

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

In a square pyramid:
- The **lateral face** is the triangle formed between the apex of the pyramid and the edge of the base.
- The **base** of the pyramid refers to the length of one side of the square base.
- The **slant height** is the height of the triangular lateral face.

Given:
- Lateral face \( = 56 \)
- Base \( = 33 \)

1. Half of the base length \( = \frac{33}{2} = 16.5 \)
2. Using the Pythagorean theorem for the triangle:
\[
\text{slant height}^2 + \left(\frac{\text{base}}{2}\right)^2 = \text{lateral face}^2
\]
\[
\text{slant height}^2 + 16.5^2 = 56^2
\]
\[
\text{slant height}^2 + 272.25 = 3136
\]
\[
\text{slant height}^2 = 3136 - 272.25
\]
\[
\text{slant height}^2 = 2863.75
\]
\[
\text{slant height} = \sqrt{2863.75} \approx 53.6
\]

As the question requests only an integer, the nearest integer is \( 54 \).

Thus, the length of the slant height of the square pyramid is approximately **54** miles.