Question

For the square pyramid, the lateral edge length, e , is 25 in., and the slant height, s , is 24 in. Find the height of the pyramid, h . Round the answer to the nearest whole number. Hint: Before finding the length of h , you will first need to find the length of r .

Answer 1

To find the height \( h \) of the square pyramid, we first need to find the length of \( r \), which is the distance from the center of the base to one of the vertices of the base.

Given:
- Lateral edge length \( e = 25 \) in.
- Slant height \( s = 24 \) in.

1. **Finding the base side length \( a \):**

The slant height \( s \) of the pyramid is the height of the triangle formed by the slant height, half of the base side length \( \frac{a}{2} \), and the height \( h \) of the pyramid.

We can use the right triangle formed by the height \( h \), half the length of the base \( \frac{a}{2} \), and the slant height \( s \):

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

2. **Finding the lateral edge length \( e \):**

The lateral edge forms another right triangle where the height \( h \), half the length of the base \( \frac{a}{2} \), and the lateral edge \( e \):

\[
e^2 = h^2 + r^2
\]

where \( r = \frac{a \sqrt{2}}{2} \) because \( r \) is the distance from the center of the square base to a vertex of the square base.

3. **Using the Pythagorean theorem:**

We have two equations:

\[
s^2 = h^2 + \left(\frac{a}{2}\right)^2 \tag{1}
\]
\[
e^2 = h^2 + r^2 \tag{2}
\]

Substituting \( r \) in equation (2):

\[
r = \frac{a \sqrt{2}}{2}
\]
So,
\[
e^2 = h^2 + \left(\frac{a \sqrt{2}}{2}\right)^2 \rightarrow e^2 = h^2 + \frac{a^2}{2} \tag{3}
\]

Now we have:

- \( s^2 = h^2 + \frac{a^2}{4} \) from equation (1).
- \( e^2 = h^2 + \frac{a^2}{2} \) from equation (3).

Setting \( s^2 \) and \( e^2 \):

Now substituting values:

\[
s = 24 \implies s^2 = 576
\]
\[
e = 25 \implies e^2 = 625
\]

From (1):

\[
576 = h^2 + \frac{a^2}{4} \tag{4}
\]

From (3):

\[
625 = h^2 + \frac{a^2}{2} \tag{5}
\]

Subtract (4) from (5):
\[
625 - 576 = (h^2 + \frac{a^2}{2}) - (h^2 + \frac{a^2}{4})
\]
\[
49 = \frac{a^2}{2} - \frac{a^2}{4}
\]
\[
49 = \frac{2a^2 - a^2}{4} = \frac{a^2}{4}
\]
\[
a^2 = 49 \times 4 = 196
\]
\[
a = \sqrt{196} = 14
\]

4. **Find \( r \):**
\[
r = \frac{a \sqrt{2}}{2} = \frac{14 \sqrt{2}}{2} = 7 \sqrt{2}
\]

5. **Using \( e^2 = h^2 + r^2 \) to find height \( h \):**

Now substitute \( r \):
\[
e^2 = h^2 + (7 \sqrt{2})^2 = h^2 + 98
\]
\[
625 = h^2 + 98
\]
\[
h^2 = 625 - 98 = 527
\]
\[
h = \sqrt{527} \approx 22.91
\]

Rounding to the nearest whole number, the height \( h \) is approximately \( 23 \) in.

**Final Answer:**
\[
\boxed{23} \text{ inches}
\]