Editing Area (section)

==Formal definition==
{{see also|Jordan measure}}
An approach to defining what is meant by "area" is through [[axiom]]s. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:<ref name=Apostol>{{cite book|last=Apostol|first=Tom|year=1967|title=Calculus|volume=I: One-Variable Calculus, with an Introduction to Linear Algebra|pages=58–59|publisher=John Wiley & Sons |isbn=9780471000051}}</ref>
* For all ''S'' in ''M'', {{nowrap|''a''(''S'') ≥ 0}}.
* If ''S'' and ''T'' are in ''M'' then so are {{nowrap|''S'' ∪ ''T''}} and {{nowrap|''S'' ∩ ''T''}}, and also {{nowrap|1=''a''(''S''∪''T'') = ''a''(''S'') + ''a''(''T'') − ''a''(''S'' ∩ ''T'')}}.
* If ''S'' and ''T'' are in ''M'' with {{nowrap|''S'' ⊆ ''T''}} then {{nowrap|''T'' − ''S''}} is in ''M'' and {{nowrap|1=''a''(''T''−''S'') = ''a''(''T'') − ''a''(''S'')}}.
* If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and {{nowrap|1=''a''(''S'') = ''a''(''T'')}}.
* Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then {{nowrap|1=''a''(''R'') = ''hk''}}.
* Let ''Q'' be a set enclosed between two step regions ''S'' and ''T''. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. {{nowrap|''S'' ⊆ ''Q'' ⊆ ''T''}}. If there is a unique number ''c'' such that {{nowrap|''a''(''S'') ≤ c ≤ ''a''(''T'')}} for all such step regions ''S'' and ''T'', then {{nowrap|1=''a''(''Q'') = ''c''}}.

It can be proved that such an area function actually exists.<ref name=Moise>{{cite book|last=Moise|first=Edwin|title=Elementary Geometry from an Advanced Standpoint|url=https://archive.org/details/elementarygeomet0000mois|url-access=registration|access-date=15 July 2012|year=1963|publisher= Addison-Wesley Pub. Co.}}</ref>