I need someone to explain this shape to me. 
The hexagon fits in the circle perfectly, the circle fits in the square perfectly.
but the hexagon doesn't fit in the square perfectly. Doesn't this defy this formula below?
a = b
b = c
a = c (what happens in that shape is a =/= b)
Excuse my ignorance as I'm not a math student myself. I just need a clear explanation.
 A: Your confusion probably comes from the fact that you write $a = b$ to mean “$a$ fits perfectly into $b$” instead of the usual “$a$ is equal to $b$”.
The ‘$=$’ symbol carries some assumptions, and one of them is that it is a transitive relation. This is exactly what you were describing: a relation ‘$\sim$’ (using a more generic symbol to avoid confusion) is called transitive if and only if $a\sim b$ and $b\sim c$ together imply $a\sim c$ (for all $a,b,c$ in the domain of interest). Equality is a transitive relation, and several other relations are transitive as well, e.g. the ordering relation ‘$<$’. Or in this case more relevant, the congruence between shapes would be a transitive relation as well. (It would even be an equivalence relation, which is a stronger requirement and one often associated with a symbol like ‘$=$’.)
But your “fits perfectly into” relation is not transitive. As you observed yourself. And there is no mathematical reason it should be. The main reason why you assumed that it might be is in my opinion because you used a symbol for this relation which usually describes a transitive relation.
