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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)

user2860452
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    It's intended as a measure of the space (territory) within, measured as a ratio to a $1$ by $1$ square (which by definition, has area $1$). – quasi Jun 03 '17 at 18:03
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    Can you tell us a little more about your mathematical background? The desires for something formal and something non-calculus are in tension with each other, and more background will help us tailor a good answer. –  Jun 03 '17 at 18:04
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    The notion of area is actually very subtle and hard to define, despite being very clear intuitively. To get a full, rigorous exposition of this concept you would need to study measure theory, usually taught in a real analysis course. Usually the construction begins with the areas of squares or rectangles, so that the formula $A = lw$ is true more-or-less by definition. – Jair Taylor Jun 03 '17 at 18:41
  • Even for professional mathematicians the notion of area can sometimes cause headaches, with things like unmeasurable sets and the infamous Banach-Tarski paradox. – Jair Taylor Jun 03 '17 at 18:45
  • What is not clear in the Wikipedia article on area? –  Jun 03 '17 at 19:37
  • Maybe it would be helpful to learn how area is handled in a modern axiomatic treatment of geometry (like Hilbert's axioms, for example). – littleO Jun 03 '17 at 19:48
  • Related (duplicate?): "Area in axiomatic geometry". – Blue Jun 03 '17 at 23:56
  • @MikeHaskel I just took Calculus 1 course. – user2860452 Jun 07 '17 at 00:42
  • @MikeHaskel This week I started Calculus 2 – user2860452 Jun 07 '17 at 00:42

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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.

quasi
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  • Are these axioms taken from an axiomatic treatment of geometry? If so, which one? – littleO Jun 03 '17 at 19:55
  • They're just the properties, as I understand them, from various courses (e.g., geometry, calculus, real analysis). They are intended for the OP, with the goal of being descriptive, without being too formal. – quasi Jun 03 '17 at 19:57
  • Looks euclidian – user2860452 Jun 07 '17 at 00:56