Why does sum of squares appear in so many mathematical applications?

Why does sum of squares appear in so many mathematical applications?

I have some little background in statistics, where in many applications
summing of squares is an important calculation. Recently, I came across a
mention that summing squares is involved in calculating Euclidian distance
(it's a connection which never occurred to me, I am not a mathematician by
any means). This got me wondering whether there is perhaps some deep
similarity between them, so I looked "summing of squares" up in wikipedia
and found this lengthy disambiguation page, much of it referring to things
I don't understand.
This makes me wonder, if something as seemingly simple and seemingly
rather arbitrary as summing squares shows up in so many places, is there
some deeper reason for this, a connection between all of them? Is there
some common problem that summing of squares is solving in all (or most, or
many) of these cases?
As I mentioned I am not a mathematician by any means, and this may be an
idiotic question. Still, I did do some searching around and didn't find
anything obviously related.