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An illustration shows a square pyramid with labeled parts. The left, front, and right faces of the pyramid and base are visible. The back face is not visible, but its position is indicated by a dashed line. The back base edge is not visible, but its position is indicated by a dashed line. The left base edge is labeled a. The back left lateral edge is labeled e. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge and is labeled r. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. The dotted slant height line is labeled s.

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 correctly

1 answer

To find the height \( h \) of the pyramid, we will first determine the length of \( r \).

The relationship between the slant height \( s \), the height \( h \), and the base edge radius \( r \) forms a right triangle. The slant height \( s \) is the hypotenuse, the height \( h \) is one leg, and \( r \) (which is half of the length of the base edge) is the other leg.

Given:

  • The lateral edge length \( e = 25 \) inches
  • The slant height \( s = 24 \) inches

Step 1: Find \( r \).

The lateral edge \( e \) can be related to the height \( h \) and the half base edge \( r \) using the Pythagorean theorem in the triangle formed by the lateral edge \( e \), the height \( h \), and the half base edge \( r \):

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

Plugging in the known value of \( e \):

\[ 25^2 = h^2 + r^2 \] \[ 625 = h^2 + r^2 \quad (1) \]

Step 2: Relate \( s \), \( h \), and \( r \).

The slant height \( s \) relates to \( h \) and \( r \) as follows:

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

Plugging in the known value of \( s \):

\[ 24^2 = h^2 + r^2 \] \[ 576 = h^2 + r^2 \quad (2) \]

Step 3: Solve the equations.

Now we have two equations:

  1. \( 625 = h^2 + r^2 \) (from the lateral edge)
  2. \( 576 = h^2 + r^2 \) (from the slant height)

Now subtract equation (2) from equation (1):

\[ 625 - 576 = (h^2 + r^2) - (h^2 + r^2) \] \[ 49 = 625 - 576 \]

This inconsistency indicates that we should instead substitute for \( r^2 \).

From (2): \[ r^2 = 576 - h^2 \]

Substituting \( r^2 \) in equation (1):

\[ 625 = h^2 + (576 - h^2) \] \[ 625 = 576 + h^2 \] \[ h^2 = 625 - 576 \] \[ h^2 = 49 \] \[ h = \sqrt{49} \] \[ h = 7 \]

So, the height of the square pyramid is:

\[ \text{Height } h = 7 \text{ inches} \]

Thus, the final answer rounded to the nearest whole number is:

7 in.