The Pythagorean Theorem in Three Dimensions Practice

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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)

Answer 1

To find the height \( h \) of the square pyramid, we need to first determine the radius \( r \) of the base of the pyramid. Since the base of the pyramid is a square, the radius \( r \) can be defined as half of the side length of the base.

1. Finding the Length of the Base Edge:

   Let's denote the length of the base edge as \( a \). Since the base is square, we know: \[ r = \frac{a}{2} \]

2. Using the Pythagorean Theorem:

   In the right triangle formed by the height \( h \), the radius \( r \), and the slant height \( s \), we apply the Pythagorean theorem: \[ s^2 = h^2 + r^2 \]

   We know:

   • The slant height \( s = 24 \) in,
   • The lateral edge length \( e = 25 \) in.

3. Finding the Height:

   Since \( e \) is the hypotenuse of another right triangle formed with \( h \) and \( r \), we can use: \[ e^2 = h^2 + r^2 \]

   We'll solve these equations step by step. First, express \( r \) in terms of \( e \) and \( h \): \[ 25^2 = h^2 + r^2 \] \[ 625 = h^2 + r^2 \] (1)

   And we already have: \[ 24^2 = h^2 + r^2 \] \[ 576 = h^2 + r^2 \] (2)

4. Setting Up the Equations:

   Now we can set the equations (1) and (2) together:

   • From equation (2), we isolate \( r^2 \): \[ r^2 = 576 - h^2 \]

   Substitute \( r^2 \) from equation (2) into equation (1): \[ 625 = h^2 + (576 - h^2) \] \[ 625 = 576 \]

5. Solving for \( h \): Now, we can subtract \( 576 \) from both sides to find \( h^2 \): \[ 625 - 576 = h^2 \] \[ 49 = h^2 \] \[ h = \sqrt{49} = 7 \] in.

So, the height of the pyramid \( h \) is 7 inches.