Area is the
quantity 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 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, 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 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 circles. Using these formulas, the area of any 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 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.
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 determinants 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 yards (yd2), square miles (mi2), and so forth.[ Algebraically, these units can be thought of as the squares 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 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, 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 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. 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, 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 polygons).
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, 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 in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt 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 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) polygon, the Cartesian coordinates (''i''=0, 1, ..., ''n''-1) of whose ''n'' vertices are known, the area is given by the surveyor's formula:
:
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:[
: (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 polygons.
Area of curved shapes
Circles
The formula for the area of a circle (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 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:
:
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:[
:
]
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 (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''. 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 ...
: (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: where ''a'', ''b'', ''c'' are the sides of the triangle, and is half of its perimeter.[ If an angle and its two included sides are given, the area is 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 . 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 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: , 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:[
:
* 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''):
: where is the curve with the greater y-value.
* An area bounded by a function expressed in polar coordinates is:[
:
* 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 ...
with endpoints is given by the line integrals:
::
: or the ''z''-component of
::
:(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
:
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
:
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
:
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 ...
: , where ''r'' is the radius of the circular base, and ''h'' is the height. That can also be rewritten as [ or where ''r'' is the radius and ''l'' is the slant height of the cone. is the base area while is the lateral surface area of the cone.][
* Cube: , where ''s'' is the length of an edge.][
* Cylinder: , where ''r'' is the radius of a base and ''h'' is the height. The can also be rewritten as , 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 ...
: , 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 ...
: , 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 ...
: , where 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 where and is a region in the xy-plane with the smooth boundary:
:
An even more general formula for the area of the graph of a parametric surface in the vector form where is a continuously differentiable vector function of is:
:
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 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,
:
and equality holds if and only if the curve is a 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
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 (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 (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 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, , is larger than that of any non-equilateral triangle.]
The ratio of the area to the square of the perimeter of an equilateral triangle, 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, 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
* 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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