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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 answer

To find the height \( h \) of the square pyramid, we can use the relationship between the height, slant height, and radius in a right triangle formed by the height \( h \), the radius \( r \) at the center of the base to the midpoint of a base edge, and the slant height \( s \).

Step 1: Find the radius \( r \)

For a square pyramid, the radius \( r \) from the center of the base to the midpoint of one of the base edges is half the base edge length. Let's denote the length of the square base edge by \( a \).

Since we do not have the length \( a \) directly, we can use the relationship between the lateral edge, height, and half the base edge length.

Step 2: Apply the Pythagorean Theorem

The lateral edge \( e \), slant height \( s \), and height \( h \) form a right triangle where:

  • The lateral edge \( e = 25 \) in is the hypotenuse.
  • The height \( h \) is one leg.
  • The slant height \( s = 24 \) in is the other leg.

Using the Pythagorean theorem:

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

From this configuration, we need to first express \( r \) in terms of \( h \) because we do not know the base edge directly.

Step 3: Find the relationship between \( r \), \( s \), and \( h \)

Using the triangle that includes the height \( h \), slant height \( s \), and radius \( r \):

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

Step 4: Substitute the values

Substituting \( s = 24 \) in:

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

Now for the lateral edge triangle:

\[ 25^2 = h^2 + (r + \frac{a}{2})^2 \] \[ 625 = h^2 + (r + \frac{a}{2})^2 \]

However, to solve for \( h \) only using known values, we notice an easier triangle we can work. Let's go back to the relationship we formed:

Step 5: Solve for \( h \) and \( r \)

Since we do not have \( r \) directly, and it turns out we have a direct triangle via a right triangle formed.

Using the expression:

From the triangle containing height \( h \) and distance \( r \):

Setting \( e^2 = h^2 + r^2 \): \[ 25^2 = h^2 + r^2 \]

Thus we do it for this:

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

We know these yield the equations and using distance \( r \):

Let’s simplify the quadratic limitation based on height \( h \):

From earlier, simplify:

  1. \( 625 = 576 + x \) hence we take
  2. \( x = 625 - 576 = 49 \)

Thus obtain:

Final Calculations

Where:

\[ h^2 = 49 \]

Taking the square root yields:

\[ h = 7 \text{ in} \]

So the height of the pyramid, rounded to the nearest whole number, is:

\[ \boxed{7} \text{ in} \]