Square Pyramid Slant Height

Stand inside a square pyramid and slice it vertically through the apex and the midpoints of two opposite base edges. The cut exposes a right triangle whose legs are the vertical height $h$ and half the base side $s/2$, with the slant height as its hypotenuse: $l = \sqrt{h^2 + (s/2)^2}$. Half the base side, not the whole side, is the detail that trips everyone up the first time.

Roofers meet this as the difference between rise and rafter length on a hip or pyramid roof, and it is why a "12-foot-tall" gazebo roof needs rafters longer than 12 feet. Note also that the slant height is not the same as the pyramid's edge length, which runs corner to apex and is longer still, being $\sqrt{h^2 + s^2/2}$. Three different lengths, three different jobs, and quoting the wrong one to a supplier is an expensive phone call.