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 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 can use the given lateral edge length \( e = 25 \) inches and the slant height \( s = 24 \) inches.

First, we can find the length of \( r \), which is half the length of the base of the pyramid. We know that in a right triangle formed by the height of the pyramid, the slant height, and half the base, the relationship can be expressed using the Pythagorean theorem.

### Step 1: Use the Pythagorean Theorem

We can set up the following equation using the Pythagorean theorem:

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

Where:
- \( e = \) lateral edge length,
- \( h = \) vertical height,
- \( r = \) half the base length,
- \( s = \) slant height.

Since we want to find \( h \), we can take another approach to find \( r \) using the slant height.

### Step 2: Find \( r \)

From the triangle formed by the slant height, the height, and half the base, we have:

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

We also know that the lateral edge forms another right triangle:

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

From this, let's first rewrite for \( r \) using \( s \):

\[
r^2 = s^2 - h^2
\]

Now we can insert this expression for \( r^2 \) in the equation for \( e^2 \):

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

Substituting values of \( s \) and \( e \) into one of the equations defined:

Applying \( s = 24 \) and \( e = 25 \):

#### Step 3: Calculate \( h \)

We first express \( r^2 \):

\[
24^2 = h^2 + r^2
\]
\[
576 = h^2 + r^2
\]

And using \( e^2 = 25^2 = 625 \):

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

Now we have two expressions:

1. \( 576 = h^2 + r^2 \)
2. \( 625 = h^2 + r^2 \)

We can equate \( r^2 \):

1. Rearranging the first gives: \( r^2 = 576 - h^2 \)
2. Rearranging the second gives: \( r^2 = 625 - h^2 \)

Setting them equal gives:

\[
576 - h^2 = 625 - h^2
\]

This results in:

\[
576 = 625 - h^2 \implies h^2 = 625 - 576 \implies h^2 = 49
\]

### Step 4: Solve for \( h \)

Taking the square root gives:

\[
h = \sqrt{49} = 7
\]

Finally, rounding to the nearest whole number:

\[
h \approx 7
\]

Thus, the height of the pyramid is approximately \( \boxed{7} \) inches.