TITLE: What does "area" really mean?
QUESTION [5 upvotes]: My professor had an interesting statement at the beginning of first year integral calculus.

What does area really mean? 

How do we know that the area of a circle is $\pi r^2$? Archimedes used the method of exhaustion.
Is there a good generalisation for the meaning of area with respect to how it concerns integral calculus? I apologise if the question is either too general or just stupid. But it's interesting, and I would appreciate more interesting examples on the nature of area in general.

REPLY [2 votes]: The traditional way that these terms gain meaning is as follows. People start with a consensus on the area of a very basic object, like a square. The trick is then to extend the definition to more general shapes in a way that (1) agrees with the old definition for basic objects, (2) satisfies nice properties. It is often the case that these two conditions entirely determine the generalized definition. For example, suppose you cut a shape into pieces: you want the sum of the area of the pieces to be the sum of the whole shape. Well, a square can be cut into triangles, so this already determines the area of a triangle. Since a polygon can always be cut into triangles, we now have defined the area for all polygons. Since circles can be approximated by polygons, this determines the area of a circle (so long as there is a property that allows us to make sense of "approximation").
In this way, a definition of area arises from an understanding of the way area should behave, and the knowledge of the area of a very basic object. For the purposes of integral calculus, you want to generalize the definition to very strange sets, so you need to think more carefully about the sort of properties you expect area to have. This leads to a subject called measure theory.