In Euclidean Geometry we definie area as (for polygons) a positive real number satisfing:
1)Two congruent polygons have the same area.
2)The area of a rectangle is the product of base and height.
3)A polygon which is a union of two polygons not having an interior point is the sum of the areas of the two polygons.
The area of a polygon is well-defined.
Meaning no matter how you disect a polygon as smaller polygons and sum them you always end up the the same real number.
This problem seems extremely difficult. It seems to be connected to graph theory (get it? connected to graph theory ). How do you prove such a monster problem.
However, if you define area not in terms of polygons but rather in terms of function. Then the problem is easier. You define it area to be the limit of the Riemann sum. And the fact is that no matter how a Riemann sum is partitioned as long as the norm is convergent to zero the limit (if exists) is unique. Thus, area is well-defined.