Cone Slant Height

Slice a right circular cone straight down through its apex and the cut face is an isosceles triangle. Half of it is a right triangle with legs $r$ and $h$ and hypotenuse $l$, so Pythagoras hands over $l = \sqrt{r^2 + h^2}$ with no further work. Every cone formula that mentions surface uses $l$; every cone formula that mentions volume uses $h$, and this is the bridge between them.

In practice you measure whichever is easier and convert. A tape laid up the outside of a silo roof gives you the slant directly. A laser measuring the peak above the eaves gives you the vertical height. Roofers, tent makers and conveyor designers all live on this conversion, and the useful sanity check is that $l$ must always exceed both $r$ and $h$. If a supplied slant height comes out shorter than the radius, someone has mixed up which measurement is which.