[KAJLogic]

Intro:
I have asked a lot of questions on this forum and I love all you responders so much for indulging me. This thread will probably end up being incredibly long, essentially what I'm trying to figure out is the validation for some of these things I think are accurate. I have seen some mathamatical conclusions that agree with some of the things I've learned, however, due to it all being conceptual I have no basis for confidence.

Question:
Like the title say's I only know through a priori thought that it seems to make mathamatical sense. But for starters, how do we actually detect electron's distance from the nucleus? What I am trying to figure out is what is the basis for our volume formulas. We must have witnessed electron distribution and began calculating but how?