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Area is the
quantity Quantity or amount 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 multiple of a unit ...
that expresses the extent of a
region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
on the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
or on a curved
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. The area of a plane region or ''plane area'' refers to the area of a
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
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 integration. Planar laminas can be use ...
, while '' surface area'' refers to the area of an
open surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
or the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
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 analogue of the length of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
(a one-dimensional concept) or the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
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 quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s 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 The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...
is defined to have area one, and the area of any other shape or surface is a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. There are several well-known formulas for the areas of simple shapes such as
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s, rectangles, and
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s. Using these formulas, the area of any
polygon In geometry, a polygon () is a plane 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, the bounding circuit, or the two to ...
can be found by dividing the polygon into triangles. For shapes with curved boundary,
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
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 () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, 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 Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
, 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 to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...
. Area plays an important role in modern mathematics. In addition to its obvious importance in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
and calculus, area is related to the definition of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 matrices ...
, and is a basic property of surfaces in differential geometry. do Carmo, Manfredo (1976). ''Differential Geometry of Curves and Surfaces''. Prentice-Hall. p. 98, In
analysis Analysis ( : analyses) is the process of breaking a complex topic or 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 Aristotle (3 ...
, the area of a subset of the plane is defined using Lebesgue measure,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 axioms. "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'', . * If ''S'' and ''T'' are in ''M'' then so are and , and also . * If ''S'' and ''T'' are in ''M'' with then is in ''M'' and . * If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and . * Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then . * 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. . If there is a unique number ''c'' such that for all such step regions ''S'' and ''T'', then . 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 units, used in every country globally. In the United States the U.S. customary uni ...
has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (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) of area, used mainly in the United States and partially in Canada, the United Kingdom, Bang ...
(ft2),
square yard The square yard (Northern India: gaj, Pakistan: gaz) is an imperial unit and U.S. customary unit of area. It is in widespread use in most of the English-speaking world, particularly the United States, United Kingdom, Canada, Pakistan and India. ...
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. Retrieved February 22, 2012. is an imperial and US unit of measure for area. One square mile is an ar ...
s (mi2), and so forth. Algebraically, these units can be thought of as the
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the corresponding length units. The SI unit of area is the square metre, which is considered an SI derived unit.


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 thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian ''millione'' (''milione'' in modern Italian), from ''mille'', "thousand", plus the au ...
square metres * 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres * 1 square centimetre =
100 100 or one hundred ( Roman numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to de ...
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 quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the conversion between the corresponding length units. :1 foot = 12 inches, the relationship between square feet and square inches is :1 square foot = 144 square inches, where 144 = 122 = 12 × 12. Similarly: * 1 square yard = 9 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 system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Interna ...
, 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 equal to a square with 100- metre sides (1 hm2), or 10,000 m2, and is primarily used in the measurement of land. There are 100 hectares in one square kilometre. An acre is ...
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 Tetrad ('group of 4') or tetrade may refer to: * Tetrad (area), an area 2 km x 2 km square * Tetrad (astronomy), four total lunar eclipses within two years * Tetrad (chromosomal formation) * Tetrad (general relativity), or frame field ** Tetra ...
, the hectad, and the myriad. The 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 A barn is an agricultural building usually on farms and used for various purposes. In North America, a barn refers to structures that house livestock, including cattle and horses, as well as equipment and fodder, and often grain.Allen G. ...
, 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 that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
. In
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
, * 20 dhurki = 1 dhur * 20 dhur = 1 khatha * 20 khata = 1 bigha * 32 khata = 1 acre


History


Circle area

In the 5th century BCE,
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
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, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex p ...
, but did not identify the
constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
. Eudoxus of Cnidus, 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 used the tools of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
to show that the area inside a circle is equal to that of a
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
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 Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
, 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 vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
s). Swiss scientist
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
in 1761 proved that π, the ratio of a circle's area to its squared radius, is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, 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 name ...
proved that π2 is irrational; this also proves that π is irrational. In 1882, German mathematician
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
proved that π is transcendental (not the solution of any
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
with rational coefficients), confirming a conjecture made by both Legendre and Euler.


Triangle area

Heron (or Hero) of Alexandria found what is known as
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
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 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 an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
, a great
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
-
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, moons, comets and galaxies – in either ...
from the classical age of Indian mathematics and
Indian astronomy Astronomy has long history in Indian subcontinent stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley civilisation or earlier. Astronomy later developed as a dis ...
, expressed the area of a triangle as one-half the base times the height in the '' Aryabhatiya'' (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 The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of th ...
"), written by
Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gau ...
.


Quadrilateral area

In the 7th century CE, Brahmagupta developed a formula, now known as
Brahmagupta's formula In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic ...
, for the area of a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
(a
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and
Karl Georg Christian von Staudt Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic. Life and influence Karl was born in the Free Imperial City of Rothenburg, which is n ...
independently found a formula, known as
Bretschneider's formula In geometry, Bretschneider's formula is the following expression for the area of a general quadrilateral: : K = \sqrt ::= \sqrt . Here, , , , are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, sinc ...
, for the area of any quadrilateral.


General polygon area

The development of Cartesian coordinates by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...
locations by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in the 19th century.


Areas determined using calculus

The development of
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas 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 ...
) polygon, the Cartesian coordinates (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. 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 or axiom. On the other hand, if
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
is developed before arithmetic, this formula can be used to define multiplication of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.


Dissection, parallelograms, and triangles

Most other simple formulas for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a
trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
and a
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
, 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, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
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 mod ...
triangles, as shown in the figure to the right. It follows that the area of each
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is half the area of the parallelogram: :A = \fracbh  (triangle). Similar arguments can be used to find area formulas for the
trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
as well as more complicated
polygon In geometry, a polygon () is a plane 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, the bounding circuit, or the two to ...
s.


Area of curved shapes


Circles

The formula for the area of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
(more properly called the area enclosed by a circle or the area of a disk) is based on a similar method. Given a circle of radius , it is possible to partition the circle into
sectors Sector may refer to: Places * Sector, West Virginia, U.S. Geometry * Circular sector, the portion of a disc enclosed by two radii and a circular arc * Hyperbolic sector, a region enclosed by two radii and a hyperbolic arc * Spherical sector, a p ...
, 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 In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
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 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 infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. In ancient times, the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
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 Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
. 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 and semi-minor axes and the formula is: :A = \pi xy .


Non-planar surface area

Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see:
developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
s). For example, if the side surface of a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
(or any
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
, 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 () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
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 in his work ''
On the Sphere and Cylinder ''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of t ...
''. 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 infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.


General formulas


Areas of 2-dimensional figures

* A
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
: \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, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
'' 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 The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian c ...
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 infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
to find the area. * A
simple polygon In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...
constructed on a grid of equal-distanced points (i.e., points with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1 ...
.


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 graphs of two functions is equal to the
integral 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 wit ...
of one
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
, ''f''(''x''), 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 In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
is: :A = \int r^2 \, d\theta. * The area enclosed by a
parametric curve In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
\vec u(t) = (x(t), y(t)) with endpoints \vec u(t_0) = \vec u(t_1) is given by the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
s: :: \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 A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...
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 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 A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
: \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 A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
: 2\pi r(r + h), where ''r'' is the radius of a base and ''h'' is the height. The 2\pi r can also be rewritten as \pi d, where ''d'' is the diameter. *
Prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
: 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 A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
: 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 In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
: 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 A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
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 A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie o ...
. The areas of irregular (and thus arbitrary) polygons can be calculated using the " Surveyor's formula" (shoelace formula).


Relation of area to perimeter

The
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
states that, for a closed curve of length ''L'' (so the region it encloses has
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...
''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 shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
. 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 In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
that is "tipped over" arbitrarily far so that two of its
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s 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 In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
(the term for the perimeter of a circle) equals half the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''r''. This can be seen from the area formula ''πr''2 and the circumference formula 2''πr''. The area of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
is half its perimeter times the
apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
(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 In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of the fractal.


Area bisectors

There are an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are
concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...
at the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
; 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 In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
). 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
chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...
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 In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
. Familiar examples include soap bubbles. The question of the 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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
(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 In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
for triangles states that the triangle of greatest area among all those with a given perimeter is
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. 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 In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
, a cyclic quadrilateral with integer sides, integer diagonals, and integer area. *
Equiareal map In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures. Properties If ''M'' and ''N'' are two Riemannian (or pseudo-Riemannian) surfaces, the ...
*
Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...
, a triangle with integer sides and integer area. *
List of triangle inequalities In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of th ...
*
One-seventh area triangle In plane geometry, a triangle ''ABC'' contains a triangle having one-seventh of the area of ''ABC'', which is formed as follows: the sides of this triangle lie on cevians ''p, q, r'' where :''p'' connects ''A'' to a point on ''BC'' that is one-thi ...
, an inner triangle with one-seventh the area of the reference triangle. :*
Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...
, a generalization of the one-seventh area triangle. *
Orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
—A list of areas by size. * Derivation of the formula of a pentagon *
Planimeter A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...
, an instrument for measuring small areas, e.g. on maps. *
Area of a convex quadrilateral Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...
*
Robbins pentagon In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers. History Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a ...
, a cyclic pentagon whose side lengths and area are all rational numbers.


References


External links

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