## Area |

Area | |
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Common symbols | A |

^{2}] | |

In | 1 ^{2} |

**Area** is the ^{[1]} It is the two-dimensional analog of the

The area of a shape can be measured by comparing the shape to ^{[2]} In the ^{2}), which is the area of a square whose sides are one ^{[3]} A shape with an area of three square metres would have the same area as three such squares. In

There are several well-known ^{[4]} For shapes with curved boundary, ^{[5]}

For a solid shape such as a ^{[1]}^{[6]}^{[7]} Formulas for the surface areas of simple shapes were computed by the

Area plays an important role in modern mathematics. In addition to its obvious importance in ^{[8]} In ^{[9]} though not every subset is measurable.^{[10]} In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.^{[1]}

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

- formal definition
- units
- history
- area formulas
- area bisectors
- optimization
- see also
- references
- external links

An approach to defining what is meant by "area" is through

- 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.^{[11]}