Question

A pyramid frustum with labeled dimensions.
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

Let the distance from the center of the base to the midpoint of a side be r. For a square pyramid the slant height s, the height h, and r satisfy
s^2 = h^2 + r^2,
and the lateral edge e, the height h, and the distance from center to a corner R satisfy
e^2 = h^2 + R^2, with R = r√2 (center-to-corner = √2·center-to-side-midpoint).

Thus
e^2 = h^2 + 2r^2 and s^2 = h^2 + r^2.
Subtracting gives r^2 = e^2 − s^2 = 25^2 − 24^2 = 625 − 576 = 49, so r = 7 in.

Then h^2 = s^2 − r^2 = 576 − 49 = 527, so h = √527 ≈ 22.96 in.

Rounded to the nearest inch: h = 23 in.