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 . (1 point) The height of the square pyramid is ____ in.

Answer 1

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

1. The lateral edge \( e \) of the pyramid is given as 25 in, and the slant height \( s \) is given as 24 in.
2. In a square pyramid, the lateral edge \( e \), the height \( h \), and the slant height \( s \) will form a right triangle with the base radius \( r \).

We can use the Pythagorean theorem in the triangle formed by \( h \), \( r \), and \( s \):

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

Since we need to find \( r \), we can use the relationship between the lateral edge \( e \), slant height \( s \), and radius \( r \):

\[ e^2 = r^2 + h^2 \quad \text{(1)} \] \[ s^2 = r^2 + h^2 \quad \text{(2)} \]

From (1), we substitute \( e = 25 \) and from (2) \( s = 24 \):

\[ 25^2 = r^2 + h^2 \quad \text{(1)} \] \[ 24^2 = r^2 + h^2 \quad \text{(2)} \]

Calculating the squares:

\[ 625 = r^2 + h^2 \quad \text{(1)} \] \[ 576 = r^2 + h^2 \quad \text{(2)} \]

Next, we can express the height \( h \) in terms of \( r \) using the slant height equation (2):

1. From equation (2):

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

We can express \( h^2 \) as:

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

2. Substitute this into equation (1):

\[ 625 = r^2 + (576 - r^2) \]

This simplifies to:

\[ 625 = 576 \]

Clearly, solving these equations directly will not yield \( r \), so we use \( s \) to determine \( r \) through the Pythagorean relation:

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

We have already:

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

Let’s also express the relationship of \( e \) as:

\[ 25^2 = r^2 + h^2 \] or: \[ 625 = r^2 + h^2 \]

From the equations, we have a contradiction because substituting every \( h \) results in the fact that we need to manipulate either directly solving or using \( r \).

Finding \( r \) specifically will yield \( h \).

Using geometry, the \( r \) can be seen as \( 12 \) using the context.

So:

Use:

1. Solve for r in terms of e to give that pyramid height needs reevaluation for \( i \):

\[ r = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49} = 7 \]

Now substituting back to find height:

\[ h = \sqrt{576 - r^2} = \sqrt{576 - 49} = \sqrt{527} \]

Calculating gives:

\[ h \approx 23 \]

Rounding-off gives \( h \):

Thus: The height of the square pyramid is \( 23 \) inches.