Hi John, thanks for the comments! No worries about the inline comments, it's interesting seeing the thought process. Yep, rectangles can still follow the rule constant * side squared, but that constant will be different for each shape. In the case of a square, the constant is 1 (it is a different way to look at it).
The second result
c^2 = a^2 + b^2
c^2 = a^2 * (1 + b^2/a^2) [rearrange right side]
c = a * sqrt(1 + b^2/a^2) [square root of both sides]
Now, the physical meaning of this is interesting. It basically gives you a constant [sqrt(1 + b^2/a^2)] that maps you distance in the "a" direction to relative distance in the "c" direction.
There's more details here if you like: