          # Area

Area
Common symbols
A
SI unitSquare metre [m2]
In SI base unitsm2

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

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

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 ST and ST, and also a(ST) = a(S) + a(T) − a(ST).
• If S and T are in M with ST then TS is in M and a(TS) = 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. SQT. 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.

Other Languages
Адыгэбзэ: ЩIыпIэ инагъ
Afrikaans: Oppervlakte
Alemannisch: Flächeninhalt
العربية: مساحة
aragonés: Aria
ܐܪܡܝܐ: ܫܛܝܚܘܬܐ
অসমীয়া: ক্ষেত্ৰফল
авар: Площадь
تۆرکجه: مساحت
Bân-lâm-gú: Biān-chek
башҡортса: Майҙан
беларуская: Плошча
беларуская (тарашкевіца)‎: Плошча
भोजपुरी: क्षेत्रफल
български: Площ
Boarisch: Flächn
bosanski: Površina
brezhoneg: Gorread
català: Àrea
Чӑвашла: Лаптăк
Cebuano: Langyab
čeština: Obsah
chiShona: Nharaunda
Cymraeg: Arwynebedd
dansk: Areal
ދިވެހިބަސް: އަކަމިން
dolnoserbski: Wopśimjeśe płoni
eesti: Pindala
Ελληνικά: Εμβαδόν
español: Área
Esperanto: Areo
euskara: Azalera
فارسی: مساحت
føroyskt: Vídd
Frysk: Oerflak
Gaelg: Eaghtyr
Gàidhlig: Farsaingeachd
galego: Área

ગુજરાતી: ક્ષેત્રફળ

한국어: 넓이
Hawaiʻi: ʻAlea
հայերեն: Մակերես
हिन्दी: क्षेत्रफल
hornjoserbsce: Wobsah přestrjenje
hrvatski: Površina
Ido: Areo
Ilokano: Kalawa
Bahasa Indonesia: Luas
interlingua: Area
Ирон: Фæзуат
íslenska: Flatarmál
italiano: Area
עברית: שטח
Jawa: Jembar
ქართული: ფართობი
Kiswahili: Eneo
kurdî: Rûerd
Кыргызча: Аянт
latviešu: Laukums
Lëtzebuergesch: Fläch
lietuvių: Plotas
Limburgs: Oppervlak
lingála: Etando
मैथिली: क्षेत्रफल
македонски: Плоштина
Malagasy: Velarantany
მარგალური: ფართობი
مازِرونی: گتی
Bahasa Melayu: Keluasan
Mìng-dĕ̤ng-ngṳ̄: Miêng-cék
Mirandés: Ária
монгол: Талбай
မြန်မာဘာသာ: ဧရိယာ
Nederlands: Oppervlakte
Nedersaksies: Oppervlakte
नेपाली: क्षेत्रफल

нохчийн: Майда
Nordfriisk: Areal (miat)
norsk: Areal
norsk nynorsk: Flatevidd
occitan: Aira
олык марий: Кумдык
oʻzbekcha/ўзбекча: Yuza
ਪੰਜਾਬੀ: ਖੇਤਰਫਲ
Pälzisch: Fläche
Patois: Ieria
ភាសាខ្មែរ: ក្រលាផ្ទៃ
Plattdüütsch: Flach
português: Área
română: Arie
русский: Площадь
संस्कृतम्: क्षेत्रफलम्
Scots: Aurie
Simple English: Area
سنڌي: ايراضي
slovenčina: Plocha (útvar)
slovenščina: Površina
словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ: Пространиѥ
Soomaaliga: Bed
کوردی: ڕووبەر
српски / srpski: Површина
srpskohrvatski / српскохрватски: Površina (geometrija)
Basa Sunda: Aréa
suomi: Pinta-ala
svenska: Area
Tagalog: Sukat
தமிழ்: பரப்பளவு
татарча/tatarça: Mäydan
తెలుగు: విస్తీర్ణం
тоҷикӣ: Масоҳат
Türkçe: Alan
українська: Площа
اردو: رقبہ
Tiếng Việt: Diện tích
Võro: Pindala
West-Vlams: Ippervlak
Winaray: Kahaluag
Wolof: Yaatuwaay

ייִדיש: שטח
Yorùbá: Ààlà

Zeêuws: Oppervlak
žemaitėška: Pluots  