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Area is the
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
that expresses the extent of a
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...
region In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the wo ...
,
shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. ...

shape
, or
planar lamina In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integral, integration. Planar laminas c ...
, in the
plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse *Plane (Magic: Th ...
.
Surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

Surface area
is its analog on the two-dimensional
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
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 Paint is any pigmented liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers to a fluid flow, flow in which the material densi ...

paint
necessary to cover the surface with a single coat. It is the two-dimensional analog of the
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...
(a one-dimensional concept) or the
volume Volume is a expressing the of enclosed by a . For example, the space that a substance (, , , or ) or occupies or contains. Volume is often quantified numerically using the , the . The volume of a container is generally understood to be the ...

volume
of a solid (a three-dimensional concept). The area of a shape can be measured by comparing the shape to
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...

square
s of a fixed size. In the
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(SI), the standard unit of area is the
square metre The square metre ( international spelling as used by the International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organis ...
(written as m2), which is the area of a square whose sides are one
metre The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English o ...
long. A shape with an area of three square metres would have the same area as three such squares. In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the
unit square 300px, The unit square in the Euclidean geometry, real plane In mathematics, a unit square is a square (geometry), square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian coordinate system#Ca ...
is defined to have area one, and the area of any other shape or surface is a
dimensionless In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. There are several well-known
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

formula
s for the areas of simple shapes such as
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

triangle
s,
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
s, and
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
s. Using these formulas, the area of any
polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region (mathematic ...

polygon
can be found by dividing the polygon into triangles. For shapes with curved boundary,
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

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 A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

sphere
, cone, or cylinder, the area of its boundary surface is called the
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

surface area
. Formulas for the surface areas of simple shapes were computed by the
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
, but computing the surface area of a more complicated shape usually requires
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...
. Area plays an important role in modern mathematics. In addition to its obvious importance in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
and calculus, area is related to the definition of
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
s in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (math ...
, and is a basic property of surfaces in
differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
. do Carmo, Manfredo (1976). ''Differential Geometry of Curves and Surfaces''. Prentice-Hall. p. 98, In
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Ari ...

analysis
, the area of a subset of the plane is defined using
Lebesgue measureIn measure theory In mathematics, a measure on a set (mathematics), set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization ...
,Walter Rudin (1966). ''Real and Complex Analysis'', McGraw-Hill, . 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
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

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: * 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.


Units

Every
unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country globally. In the United States the U.S. c ...
has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in
square metre The square metre ( international spelling as used by the International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organis ...
s (m2), square centimetres (cm2), square millimetres (mm2),
square kilometre Square kilometre (American and British English spelling differences#-re, -er, International spelling as used by the International Bureau of Weights and Measures) or square kilometer (American spelling), symbol km2, is a multiple of the square metre, ...
s (km2),
square feet The square foot (plural square feet; abbreviated sq. ft, sf, or ft2; also denoted by '2) is an imperial unit and U.S. customary unit (non-SI, non-metric system, metric) of area, used mainly in the United States and partially in Canada, the United Ki ...
(ft2),
square yard The square yard (Northern India North India is a loosely defined region consisting of the northern part of India India (Hindi: ), officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and ...
s (yd2),
square mile The square mile (abbreviated as sq mi and sometimes as mi2)Rowlett, Russ (September 1, 2004) University of North Carolina at Chapel Hill The University of North Carolina at Chapel Hill (UNC, UNC-Chapel Hill, North Carolina, Chapel Hill, o ...
s (mi2), and so forth. Algebraically, these units can be thought of as the
squares In geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in ...
of the corresponding length units. The SI unit of area is the square metre, which is considered an
SI derived unit SI derived units are units of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects o ...
.


Conversions

Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m2 and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are: * 1 square kilometre =
1,000,000 One million (1,000,000), or one thousand 1000 or one thousand is the natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is th ...

1,000,000
square metres * 1 square metre =
10,000 10,000 (ten thousand) is the natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are '' ...
square centimetres = 1,000,000 square millimetres * 1 square centimetre =
100 100 or one hundred (Roman numeral Roman numerals are a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from L ...
square millimetres.


Non-metric units

In non-metric units, the conversion between two square units is the
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
of the conversion between the corresponding length units. :1
foot The foot (plural: feet) is an anatomical Anatomy (Greek ''anatomē'', 'dissection') is the branch of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemis ...
= 12
inch Measuring tape with inches The inch (symbol: in or ″) is a unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television s ...
es, the relationship between square feet and square inches is :1 square foot = 144 square inches, where 144 = 122 = 12 × 12. Similarly: * 1 square yard = square feet * 1 square mile = 3,097,600 square yards = 27,878,400 square feet In addition, conversion factors include: * 1 square inch = 6.4516 square centimetres * 1 square foot = square metres * 1 square yard = square metres * 1 square mile = square kilometres


Other units including historical

There are several other common units for area. The are was the original unit of area in the
metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culminated in the definition of the (SI), under the oversight of an international stan ...

metric system
, with: * 1 are = 100 square metres Though the are has fallen out of use, the
hectare The hectare (; SI symbol: ha) is a non-SI metric unit of area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface ...

hectare
is still commonly used to measure land: Chapter 5. * 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres Other uncommon metric units of area include the
tetrad A tetrad is a "group of four". Tetrad or tetrade may also refer to: * a tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
, the
hectad A hectad is an area 10 km x 10 km square. The term has a particular use in connection with the British Ordnance Survey British national grid reference system, national grid, and then refers to any of the 100 such squares which make up a s ...
, and the
myriad A myriad (from Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: M ...
. The
acre The acre is a of land area used in the and systems. It is traditionally defined as the area of one by one (66 by 660 feet), which is exactly equal to 10 square chains, of a square mile, or 43,560 square feet, and approximately 4,047 m ...

acre
is also commonly used to measure land areas, where * 1 acre = 4,840 square yards = 43,560 square feet. An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of
barns in Lubbock, Texas, U.S., was used as a teaching facility until 1967. , Coggeshall, England, originally part of the Cistercian monastery of Coggeshall. Dendrochronologically dated from 1237–1269, it was restored in the 1980s by the Coggeshall ...
, such that: * 1 barn = 10−28 square meters. The barn is commonly used in describing the cross-sectional area of interaction in
nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ent ...
. In
India India, officially the Republic of India (: ), is a country in . It is the by area, the country, and the most populous in the world. Bounded by the on the south, the on the southwest, and the on the southeast, it shares land borders wit ...

India
, * 20 dhurki = 1 dhur * 20 dhur = 1 khatha * 20 khata = 1
bigha The bigha (also formerly beegah) is a traditional unit of measurement of area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensio ...
* 32 khata = 1 acre


History


Circle area

In the 5th century BCE,
Hippocrates of Chios Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), ...
was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the
lune of Hippocrates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

lune of Hippocrates
, but did not identify the
constant of proportionality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
.
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from ...
, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's ''Elements'' dealt with equality of areas between two-dimensional figures. The mathematician
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
used the tools of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
to show that the area inside a circle is equal to that of a
right triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Eng ...

right triangle
whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book ''
Measurement of a Circle ''Measurement of a Circle'' or ''Dimension of the Circle'' ( Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Propositions Prop ...
''. (The circumference is 2''r'', and the area of a triangle is half the base times the height, yielding the area ''r''2 for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regul ...

hexagon
, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with
circumscribed polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertex (geometry), vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. ...

circumscribed polygon
s). Swiss scientist
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French language, French; 26 or 28 August 1728 – 25 September 1777) was a Switzerland, Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics ...
in 1761 proved that , the ratio of a circle's area to its squared radius, is
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
, meaning it is not equal to the quotient of any two whole numbers. English translation by Catriona and David Lischka. In 1794 French mathematician
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named a ...
proved that π2 is irrational; this also proves that π is irrational. In 1882, German mathematician
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
proved that π is transcendental (not the solution of any
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
with rational coefficients), confirming a conjecture made by both Legendre and Euler.


Triangle area

found what is known as
Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

Heron's formula
for the area of a triangle in terms of its sides, and a proof can be found in his book, ''Metrica'', written around 60 CE. It has been suggested that
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
knew the formula over two centuries earlier, and since ''Metrica'' is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 499
Aryabhata Aryabhata (, ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, a great
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
-
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

astronomer
from the classical age of
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
and Indian astronomy, expressed the area of a triangle as one-half the base times the height in the ''
Aryabhatiya ''Aryabhatiya'' (IAST The International Alphabet of Sanskrit Transliteration (IAST) is a transliteration scheme that allows the lossless romanisation of Brahmic family, Indic scripts as employed by Sanskrit and related Indic languages. It is ...
'' (section 2.6). A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in ''Shushu Jiuzhang'' (" Mathematical Treatise in Nine Sections"), written by
Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the Sinosphere, including China, Japan, Ko ...
.


Quadrilateral area

In the 7th century CE,
Brahmagupta Brahmagupta ( – ) was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

Brahmagupta
developed a formula, now known as
Brahmagupta's formula In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; ...
, for the area of a
cyclic quadrilateral In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

cyclic quadrilateral
(a
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and ...

quadrilateral
inscribed{{unreferenced, date=August 2012 Image:Inscribed circles.svg, frame, Inscribed circles of various polygons image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid ...
in a circle) in terms of its sides. In 1842 the German mathematicians
Carl Anton BretschneiderCarl Anton Bretschneider (27 May 1808 – 6 November 1878) was a mathematician from Gotha Gotha () is the fifth-largest city in Thuringia Thuringia (; german: Thüringen ), officially the Free State of Thuringia ( ), is a states of Germany, sta ...
and
Karl Georg Christian von Staudt Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
independently found a formula, known as
Bretschneider's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

Bretschneider's formula
, for the area of any quadrilateral.


General polygon area

The development of
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...
by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
locations by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
in the 19th century.


Areas determined using calculus

The development of
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Deriv ...
in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
and the
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

surface area
s of various curved three-dimensional objects.


Area formulas


Polygon formulas

For a non-self-intersecting (
simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

simple
) polygon, the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...
(x_i, y_i) (''i''=0, 1, ..., ''n''-1) of whose ''n'' vertices are known, the area is given by the surveyor's formula: :A = \frac \Biggl\vert \sum_^( x_i y_ - x_ y_i) \Biggr\vert where when ''i''=''n''-1, then ''i''+1 is expressed as modulus ''n'' and so refers to 0.


Rectangles

The most basic area formula is the formula for the area of a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
. Given a rectangle with length and width , the formula for the area is: :  (rectangle). That is, the area of the rectangle is the length multiplied by the width. As a special case, as in the case of a square, the area of a square with side length is given by the formula: :  (square). The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a
definition A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical ...

definition
or
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

axiom
. On the other hand, if
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
is developed before
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
, this formula can be used to define
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
of
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s.


Dissection, parallelograms, and triangles

Most other simple formulas for area follow from the method of
dissection Dissection (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
. This involves cutting a shape into pieces, whose areas must to the area of the original shape. For an example, any
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
can be subdivided into a
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

trapezoid
and a
right triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Eng ...

right triangle
, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: :  (parallelogram). However, the same parallelogram can also be cut along a
diagonal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

diagonal
into two
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
triangles, as shown in the figure to the right. It follows that the area of each
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

triangle
is half the area of the parallelogram: :A = \fracbh  (triangle). Similar arguments can be used to find area formulas for the
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

trapezoid
as well as more complicated
polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region (mathematic ...

polygon
s.


Area of curved shapes


Circles

The formula for the area of a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
(more properly called the area enclosed by a circle or the area of a
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
) is based on a similar method. Given a circle of radius , it is possible to partition the circle into Circular sector, sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is : :  (circle). Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit (mathematics), limit of the areas of the approximate parallelograms is exactly , which is the area of the circle. This argument is actually a simple application of the ideas of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

calculus
. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Deriv ...
. Using modern methods, the area of a circle can be computed using a definite integral: :A \;=\;2\int_^r \sqrt\,dx \;=\; \pi r^2.


Ellipses

The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major axis, semi-major and semi-minor axis, semi-minor axes and the formula is: :A = \pi xy .


Surface area

Most basic formulas for
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

surface area
can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (geometry), cylinder (or any prism (geometry), prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone (geometry), cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

sphere
is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
in his work ''On the Sphere and Cylinder''. The formula is: :  (sphere), where is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

calculus
.


General formulas


Areas of 2-dimensional figures

* A
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

triangle
: \tfrac12Bh (where ''B'' is any side, and ''h'' is the distance from the line on which ''B'' lies to the other vertex of the triangle). This formula can be used if the height ''h'' is known. If the lengths of the three sides are known then ''
Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

Heron's formula
'' can be used: \sqrt where ''a'', ''b'', ''c'' are the sides of the triangle, and s = \tfrac12(a + b + c) is half of its perimeter. If an angle and its two included sides are given, the area is \tfrac12 a b \sin(C) where is the given angle and and are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of \tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3). This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points ''(x1,y1)'', ''(x2,y2)'', and ''(x3,y3)''. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

calculus
to find the area. * A simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: i + \frac - 1, where ''i'' is the number of grid points inside the polygon and ''b'' is the number of boundary points. This result is known as Pick's theorem.


Area in calculus

* The area between a positive-valued curve and the horizontal axis, measured between two values ''a'' and ''b'' (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from ''a'' to ''b'' of the function that represents the curve: : A = \int_a^ f(x) \, dx. * The area between the graph of a function, graphs of two functions is equality (mathematics), equal to the integral of one function (mathematics), function, ''f''(''x''), subtraction, minus the integral of the other function, ''g''(''x''): : A = \int_a^ ( f(x) - g(x) ) \, dx, where f(x) is the curve with the greater y-value. * An area bounded by a function r = r(\theta) expressed in polar coordinates is: :A = \int r^2 \, d\theta. * The area enclosed by a parametric curve \vec u(t) = (x(t), y(t)) with endpoints \vec u(t_0) = \vec u(t_1) is given by the line integrals: :: \oint_^ x \dot y \, dt = - \oint_^ y \dot x \, dt = \oint_^ (x \dot y - y \dot x) \, dt : or the ''z''-component of :: \oint_^ \vec u \times \dot \, dt. :(For details, see .) This is the principle of the planimeter mechanical device.


Bounded area between two quadratic functions

To find the bounded area between two quadratic functions, we subtract one from the other to write the difference as :f(x)-g(x)=ax^2+bx+c=a(x-\alpha)(x-\beta) where ''f''(''x'') is the quadratic upper bound and ''g''(''x'') is the quadratic lower bound. Define the discriminant of ''f''(''x'')-''g''(''x'') as :\Delta=b^2-4ac. By simplifying the integral formula between the graphs of two functions (as given in the section above) and using Vieta's formulas, Vieta's formula, we can obtain :A=\frac=\frac(\beta-\alpha)^3,\qquad a\neq0. The above remains valid if one of the bounding functions is linear instead of quadratic.


Surface area of 3-dimensional figures

* Cone: \pi r\left(r + \sqrt\right), where ''r'' is the radius of the circular base, and ''h'' is the height. That can also be rewritten as \pi r^2 + \pi r l or \pi r (r + l) \,\! where ''r'' is the radius and ''l'' is the slant height of the cone. \pi r^2 is the base area while \pi r l is the lateral surface area of the cone. * cube: 6s^2, where ''s'' is the length of an edge. * cylinder: 2\pi r(r + h), where ''r'' is the radius of a base and ''h'' is the height. The ''2\pir'' can also be rewritten as ''\pi d'', where ''d'' is the diameter. * Prism (geometry), prism: 2B + Ph, where ''B'' is the area of a base, ''P'' is the perimeter of a base, and ''h'' is the height of the prism. * pyramid (geometry), pyramid: B + \frac, where ''B'' is the area of the base, ''P'' is the perimeter of the base, and ''L'' is the length of the slant. * rectangular prism: 2 (\ell w + \ell h + w h), where \ell is the length, ''w'' is the width, and ''h'' is the height.


General formula for surface area

The general formula for the surface area of the graph of a continuously differentiable function z=f(x,y), where (x,y)\in D\subset\mathbb^2 and D is a region in the xy-plane with the smooth boundary: : A=\iint_D\sqrt\,dx\,dy. An even more general formula for the area of the graph of a parametric surface in the vector form \mathbf=\mathbf(u,v), where \mathbf is a continuously differentiable vector function of (u,v)\in D\subset\mathbb^2 is: : A=\iint_D \left, \frac\times\frac\\,du\,dv.


List of formulas

The above calculations show how to find the areas of many common shapes. The areas of irregular polygons can be calculated using the "Surveyor's formula".


Relation of area to perimeter

The isoperimetric inequality states that, for a closed curve of length ''L'' (so the region it encloses has perimeter ''L'') and for area ''A'' of the region that it encloses, :4\pi A \le L^2, and equality holds if and only if the curve is a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
. Thus a circle has the largest area of any closed figure with a given perimeter. At the other extreme, a figure with given perimeter ''L'' could have an arbitrarily small area, as illustrated by a rhombus that is "tipped over" arbitrarily far so that two of its angles are arbitrarily close to 0° and the other two are arbitrarily close to 180°. For a circle, the ratio of the area to the circumference (the term for the perimeter of a circle) equals half the radius ''r''. This can be seen from the area formula ''πr''2 and the circumference formula 2''πr''. The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side).


Fractals

Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a fractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the fractal dimension of the fractal.


Area bisectors

There are an infinitude of lines that bisect the area of a triangle. Three of them are the Median (triangle), medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are Concurrent lines, concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. Any line through the midpoint of a parallelogram bisects the area. All area bisectors of a circle or other ellipse go through the center, and any Chord (geometry), chords through the center bisect the area. In the case of a circle they are the diameters of the circle.


Optimization

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles. The question of the filling area conjecture, filling area of the Riemannian circle remains open. The circle has the largest area of any two-dimensional object having the same perimeter. A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, \frac, is larger than that of any non-equilateral triangle. The ratio of the area to the square of the perimeter of an equilateral triangle, \frac, is larger than that for any other triangle.Chakerian, G.D. (1979) "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums''. R. Honsberger (ed.). Washington, DC: Mathematical Association of America, p. 147.


See also

* Brahmagupta quadrilateral, a cyclic quadrilateral with integer sides, integer diagonals, and integer area. * Equiareal map * Heronian triangle, a triangle with integer sides and integer area. * List of triangle inequalities#Area, List of triangle inequalities * One-seventh area triangle, an inner triangle with one-seventh the area of the reference triangle. :*Routh's theorem, a generalization of the one-seventh area triangle. * Orders of magnitude (area), Orders of magnitude—A list of areas by size. * Pentagon#Derivation of the area formula, Derivation of the formula of a pentagon * Planimeter, an instrument for measuring small areas, e.g. on maps. * Quadrilateral#Area of a convex quadrilateral, Area of a convex quadrilateral * Robbins pentagon, a cyclic pentagon whose side lengths and area are all rational numbers.


References


External links

{{Authority control Area,