What's area in geometry? I've searched in a few basic fonts but I didn't found anything formal, can you give me the most formal definition of area ( please don't explain using integral calculus... I want to know what's area in the most basic way... what's exactally the area of a square)
 A: If a planar region has an area, the area is a nonnegative real number, intended as a measure of the space (territory) within, measured as a ratio to the area of a $1$ by $1$ square (which by definition, has area $1$).

Note: We don't require all subsets of the plane to have areas (in fact, not all of them do).

The concept of area can be best understood by its properties . . .


*

*The area of the empty set is zero.

*More generally, the area of any finite set is zero.

*The area of a $1$ by $1$ square is $1$.


Let $S$ be a planar region such that the area of $S$ exists.


*

*The area of $S$ is a nonnegative real number.

*If the area of $S$ is zero, then every subset of $S$ has area zero.

*If $S$ is translated (shifted) in the plane by a fixed amount in some direction, the area remains the same.

*If $S$ is rotated about a point in the plane by an arbitrary angle, the area remains the same.

*If $S$ is reflected over a line in the plane, the area remains the same.

*If $S$ is magnified by a positive real factor $x$, the new area is $x^2$ times the old area.


Let $S$ be a planar region which may or may not have an area (i.e., the area of $S$ may be undefined).


*

*If $S$ is partitioned into two non-overlapping subregions $A,B$, and if at least two of $A,B,S$ has an area, then they all have areas, and the area of $S$ is equal to the sum of the areas of $A,B$.  

*Moreover, if $S$ is partitioned into a countable (finite or countably infinite) number of non-overlapping subregions, such that each subregion has an area, then $S$ has an area, and the area of $S$ is equal to the sum of the areas of the subregions.


While the properties listed above don't qualify as a formal definition, they serve as a set of requirements for the concept.
