**Formal Definition**

An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:

- For all
*S*in*M*,*a*(*S*) ≥ 0. - If
*S*and*T*are in*M*then so are*S*∪*T*and*S*∩*T*, and also*a*(*S*∪*T*) =*a*(*S*) +*a*(*T*) −*a*(*S*∩*T*). - If
*S*and*T*are in*M*with*S*⊆*T*then*T*−*S*is in*M*and*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*a*(*S*) =*a*(*T*). - Every rectangle
*R*is in*M*. If the rectangle has length*h*and breadth*k*then*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.*S*⊆*Q*⊆*T*. If there is a unique number*c*such that*a*(*S*) ≤ c ≤*a*(*T*) for all such step regions*S*and*T*, then*a*(*Q*) =*c*.

It can be proved that such an area function actually exists.

Read more about this topic: Area

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—Ralph Waldo Emerson (1803–1882)

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—Edgar Lee Masters (1869–1950)