While there is a naive idea of area in terms of number of unit squares that can fit inside a region, this is not the best definition. It is better to work with oriented triangles and maintain linearity.This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.