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: \[ \boxed{23} \text{ inches} \]