All the "elementary derivations of that type" are cheating.

The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating). You have to say that there is a additive area-functional on the set of all polygons. Then you probably want to prove that this functional is unique (or include it in the same axiom).

It remains to notice that your functional $A$ satisfies the same properties and nothing left to prove.

All the "elementary derivations of that type" are cheating (it may look nice but it proves nothing).

The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating). You have to say that there is a additive area-functional on the set of all polygons. Then you probably want to prove that this functional is unique (or include it in the same axiom).

It remains to notice that your functional $A$ satisfies the same properties and nothing left to prove.

Most of elementary derivations of that type are cheating.

The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating). You have to say that there is a additive area-functional on the set of all polygons. Then you probably want to prove that this functional is unique (or include it in the same axiom).

But then you notice that your functional $A$ satisfies all the same properties and there is nothing to prove.

All the "elementary derivations of that type" are cheating.

The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating). You have to say that there is a additive area-functional on the set of all polygons. Then you probably want to prove that this functional is unique (or include it in the same axiom).

It remains to notice that your functional $A$ satisfies the same properties and nothing left to prove.