TheInfoList

Volume is a scalar quantity expressing the
amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ...
of
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
enclosed by a
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
. For example, the space that a substance (
solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied ...

,
liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers to a fluid flow, flow in which the material density is constant within a fluid par ...

,
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

, or
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood plasma ...
) or
3D shape In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
occupies or contains. Volume is often quantified numerically using the
SI derived unit SI derived units are units of measurement derived from the seven SI base unit, base units specified by the International System of Units (SI). They are either dimensionless quantity, dimensionless or can be expressed as a product of one or more o ...
, the
cubic metre The cubic metre (in Commonwealth English The use of the English language English is a of the , originally spoken by the inhabitants of . It is named after the , one of the ancient that migrated from , a peninsula on the (not to b ...
. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
(gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

s. Volumes of complicated shapes can be calculated with
integral calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
if a formula exists for the shape's boundary. One-dimensional figures (such as
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...
) and
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...
shapes (such as
squares In geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in ...

) are assigned zero volume in the three-dimensional space. The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not
additive Additive may refer to: Mathematics * Additive function In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer ''n'' such that whenever ''a'' and ''b'' are coprime, the function of the product is the ...
. In ''
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
'', volume is expressed by means of the
volume formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, and is an important global Riemannian invariant. In ''
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

'', volume is a fundamental parameter, and is a
conjugate variable Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual (mathematics), duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an unc ...
to
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

.

# Units

Any unit of
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

gives a corresponding unit of volume: the volume of a
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

whose sides have the given length. For example, a
cubic centimetre A cubic centimetre (or cubic centimeter in US English) (SI unit symbol: cm3; non-SI abbreviations: cc and ccm) is a commonly used unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction t ...
(cm3) is the volume of a cube whose sides are one
centimetre 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. A centimetre (international spelling) or centimeter (American spellin ...
(1 cm) in length. In the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI), the standard unit of volume is the cubic metre (m3). The
metric system The metric system is a system of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purpose ...

also includes the
litre The litre (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar and ...

(L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus :1 = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres, so :1 cubic metre = 1000 litres. Small amounts of liquid are often measured in
millilitre The litre (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar and ...
s, where :1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where :1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are also in use, including the
cubic inch The cubic inch (symbol in3) is a unit of volume in the Imperial units and United States customary units systems. It is the volume of a cube with each of its three dimensions (length, width, and depth) being one inch long which is equivalent to 1 ...
, the
cubic foot The cubic foot (symbol ft3 or cu ft)
, .
is an Imperial unit, imperial and United States customary unit ...
, the
cubic yard A cubic yard (symbol yd3)IEEE Std 260.1-2004 is an Imperial unit, Imperial / U.S. customary unit, U.S. customary (non-SI non-Metric system, metric) unit of volume, used in the United States. It is defined as the volume of a cube with sides of 1 yard ...
, the
cubic mile Cubic may refer to: Science and mathematics * Cube (algebra) In arithmetic and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathemat ...
, the
teaspoon A teaspoon (tsp.) is an item of cutlery Cutlery includes any hand implement used in preparing, serving, and especially eating food Food is any substance consumed to provide Nutrient, nutritional support for an organism. Food is usually o ...

, the
tablespoon A tablespoon is a large spoon A spoon is a utensil consisting of a small shallow bowl (also known as a head), oval or round, at the end of a handle. A type of cutlery (sometimes called flatware in the United States), especially as part of a ta ...
, the
fluid ounce A fluid ounce (abbreviated fl oz, fl. oz. or oz. fl., old forms ℥, fl ℥, f℥, ƒ ℥) is a unit of volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is ...
, the
fluid dram The dram (alternative British spelling drachm; apothecary symbol ʒ or ℨ; abbreviated dr) Earlier version first published in '' New English Dictionary'', 1897.National Institute of Standards and Technology (October 2011). Butcher, Tina; Cook, St ...
, the
gill A gill () is a respiratory organ that many aquatic Aquatic means relating to water Water (chemical formula H2O) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent ...
, the
pint The pint (, ; symbol pt, sometimes abbreviated as ''p'') is a unit of volume or capacity in both the imperial unit, imperial and United States customary units, United States customary measurement systems. In both of those systems it is tradition ...
, the
quart The quart (symbol: qt) is an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventu ...

, the
gallon The gallon is a unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete pie ...
, the minim, the
barrel A barrel or cask is a hollow cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), ...
, the
cord Cord or CORD may refer to: * Cord (unit), a unit of measurement for firewood and pulpwood used in North America * Electrical cable, in electronics ** Extension cord ** Power cord * String (structure) made of multiple strands twisted together or ** t ...
, the
peck A peck is an imperial Imperial is that which relates to an empire, emperor, or imperialism. Imperial or The Imperial may also refer to: Places United States * Imperial, California * Imperial, Missouri * Imperial, Nebraska * Imperial, Penns ...

, the
bushel A bushel (abbreviation: bsh. or bu.) is an imperial Imperial is that which relates to an empire, emperor, or imperialism. Imperial or The Imperial may also refer to: Places United States * Imperial, California * Imperial, Missouri * Imperi ...
, the
hogshead A hogshead (abbreviated "hhd", plural "hhds") is a large cask A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wood or metal hoops. T ...
, the
acre-foot The acre-foot is a non- SI unit of volume Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance ( solid, liquid, gas, or plasma) or shape occupies or contains. Volume is of ...
and the
board foot The board foot or board-foot is a unit of measurement A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same ...
. These are all units of volume.

# Related terms

''Capacity'' is defined by the
Oxford English Dictionary The ''Oxford English Dictionary'' (''OED'') is the principal historical dictionary A historical dictionary or dictionary on historical principles is a dictionary which deals not only with the latterday meanings of words but also the historica ...
as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them". (The word ''capacity'' has other unrelated meanings, as in e.g.
capacity management Capacity management's goal is to ensure that information technology resources are sufficient to meet upcoming business requirements cost-effictively. One common interpretation of capacity management is described in the ITIL framework. ITIL versio ...
.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the litre and its derived units, and Imperial units such as
gill A gill () is a respiratory organ that many aquatic Aquatic means relating to water Water (chemical formula H2O) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent ...
,
pint The pint (, ; symbol pt, sometimes abbreviated as ''p'') is a unit of volume or capacity in both the imperial unit, imperial and United States customary units, United States customary measurement systems. In both of those systems it is tradition ...
,
gallon The gallon is a unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete pie ...
, and others. Units of volume are the cubes of
units of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country globally. In the United States the U.S. ...
. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle's fuel tank is rarely stated in cubic feet, for example, but in gallons (an imperial gallon fills a volume with 0.1605 cu ft). The ''
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

'' of an object is defined as the ratio of the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
to the volume. The inverse of density is ''
specific volumeIn thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantiti ...
'' which is defined as volume divided by mass. Specific volume is a concept important in
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

where the volume of a working fluid is often an important parameter of a system being studied. The
volumetric flow rate In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
in
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second 3 s−1. ''Volumetric space'' is a 3D
region In geography Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The ...
having a
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

in addition to capacity or volume.

# Calculus

In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the volume of a
region In geography Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The ...
''D'' in R3 is given by a
triple integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of the constant
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
$f\left(x,y,z\right)=1$ over the region and is usually written as: :$\iiint\limits_D 1 \,dx\,dy\,dz.$ In
cylindrical coordinates 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ...

, the volume integral is :$\iiint\limits_D r\,dr\,d\theta\,dz,$ In
spherical coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

(using the convention for angles with $\theta$ as the azimuth and $\varphi$ measured from the polar axis; see more on ), the volume integral is :$\iiint\limits_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .$

# Formulas

## Ratios for a cone, sphere and cylinder of the same radius and height

The above formulas can be used to show that the volumes of a
cone A cone is a three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

, sphere and
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

of the same radius and height are in the ratio 1 : 2 : 3, as follows. Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere), then the volume of the cone is :$\frac \pi r^2 h = \frac \pi r^2 \left\left(2r\right\right) = \left\left(\frac \pi r^3\right\right) \times 1,$ the volume of the sphere is :$\frac \pi r^3 = \left\left(\frac \pi r^3\right\right) \times 2,$ while the volume of the cylinder is :$\pi r^2 h = \pi r^2 \left(2r\right) = \left\left(\frac \pi r^3\right\right) \times 3.$ The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

.

## Formula derivations

### Sphere

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

is the
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of an infinite number of infinitesimally small circular disks of thickness ''dx''. The calculation for the volume of a sphere with center 0 and radius ''r'' is as follows. The surface area of the circular disk is $\pi r^2$. The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is : $y = \sqrt$ or : $z = \sqrt$ where y or z can be taken to represent the radius of a disk at a particular x value. Using y as the disk radius, the volume of the sphere can be calculated as : $\int_^r \pi y^2 \,dx = \int_^r \pi\left\left(r^2 - x^2\right\right) \,dx.$ Now : $\int_^r \pi r^2\,dx - \int_^r \pi x^2\,dx = \pi \left\left(r^3 + r^3\right\right) - \frac\left\left(r^3 + r^3\right\right) = 2\pi r^3 - \frac.$ Combining yields $V = \frac\pi r^3.$ This formula can be derived more quickly using the formula for the sphere's
surface area The surface area of a solid Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of ...

, which is $4\pi r^2$. The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to : $\int_0^r 4\pi r^2 \,dr = \frac\pi r^3.$

### Cone

The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well. However, using calculus, the volume of a
cone A cone is a three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

is the
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of an infinite number of infinitesimally thin circular disks of thickness ''dx''. The calculation for the volume of a cone of height ''h'', whose base is centered at (0, 0, 0) with radius ''r'', is as follows. The radius of each circular disk is ''r'' if ''x'' = 0 and 0 if ''x'' = ''h'', and varying linearly in between—that is, : $r \frac.$ The surface area of the circular disk is then : $\pi \left\left(r\frac\right\right)^2 = \pi r^2\frac.$ The volume of the cone can then be calculated as : $\int_0^h \pi r^2\frac dx,$ and after extraction of the constants : $\frac \int_0^h \left(h - x\right)^2 dx$ Integrating gives us : $\frac\left\left(\frac\right\right) = \frac\pi r^2 h.$

# Differential geometry

In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a volume form on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
is a
differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...
. Integrating the volume form gives the volume of the manifold according to that form. An
oriented is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
pseudo-Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...
has a natural volume form. In
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow t ...
, it can be expressed as :$\omega = \sqrt \, dx^1 \wedge \dots \wedge dx^n ,$ where the $dx^i$ are
1-form 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 an ...
s that form a positively oriented basis for the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
of the manifold, and $g$ is the
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of the matrix representation of the
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
on the manifold in terms of the same basis.

# Thermodynamics

In
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

, the volume of a
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...
is an important extensive parameter for describing its
thermodynamic state For thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governe ...
. The specific volume, an
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
, is the system's volume per unit of mass. Volume is a
function of state In the Thermodynamics#Equilibrium_thermodynamics, thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system relating several state variables or state quantities that depends only on the ...
and is interdependent with other thermodynamic properties such as
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

. For example, volume is related to the
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

of an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
by the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

.

# Computation

The task of numerically computing the volume of objects is studied in the field of
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
in computer science, investigating efficient
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s to perform this computation, approximately or exactly, for various types of objects. For instance, the convex volume approximation technique shows how to approximate the volume of any convex body using a oracle machine, membership oracle.