Question

A circus tent is the shape of a square pyramid. It has a base edge of 20meter, a slant height of 13.5m. Find the height of the tent.
Round to nearest tenth.

Answer 1

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

First, we'll denote:
- The height of the tent (pyramid) as \( h \).
- The base edge of the pyramid as \( s = 20 \) meters.
- The slant height as \( l = 13.5 \) meters.

In a square pyramid, the vertical height \( h \), the slant height \( l \), and half the length of the base edge \( \frac{s}{2} \) form a right triangle. The relationship can be expressed as follows:

\[
l^2 = h^2 + \left(\frac{s}{2}\right)^2
\]

First, calculate \( \frac{s}{2} \):

\[
\frac{s}{2} = \frac{20}{2} = 10 \text{ meters}
\]

Now substitute \( l \) and \( \frac{s}{2} \) into the Pythagorean theorem equation:

\[
(13.5)^2 = h^2 + (10)^2
\]

Calculating \( (10)^2 \) and \( (13.5)^2 \):

\[
(10)^2 = 100
\]
\[
(13.5)^2 = 182.25
\]

Substituting back into the equation:

\[
182.25 = h^2 + 100
\]

Now, isolate \( h^2 \):

\[
h^2 = 182.25 - 100 = 82.25
\]

Taking the square root to find \( h \):

\[
h = \sqrt{82.25} \approx 9.1 \text{ meters}
\]

Thus, the height of the tent is approximately **9.1 meters** when rounded to the nearest tenth.