volume

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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). The definition of (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic s. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal volume is the hypervolume.

# History

## Ancient history

The precision of volume measurements in the ancient period usually ranges between . The earliest evidence of volume calculation came from
ancient Egypt Ancient Egypt was a civilization in Northeast Africa situated in the Nile Valley. Ancient Egyptian civilization followed prehistoric Egypt and coalesced around 3100Anno Domini, BC (according to conventional Egyptian chronology) with the ...

and
Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...

as mathematical problems, approximating volume of simple shapes such as s, cylinders, frustum and s. These math problems have been written in the Moscow Mathematical Papyrus (c. 1820 BCE). In the Reisner Papyrus, ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the cubit, palm, digit) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). The last three books of Euclid's ''Elements'', written in around 300 BCE, detailed the exact formulas for calculating the volume of parallelepipeds, cones, s, cylinders, and s. The formula were determined by prior mathematicians by using a primitive form of integration, by breaking the shapes into smaller and simpler pieces. A century later, () devised approximate volume formula of several shapes used the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the and
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...

. Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a hydrostatic balance. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a submerged underwater, which will tip accordingly due to the Archimedes' principle.

## Calculus and standardization of units

In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the Post-classical, post-classical period of World history (field), global history. It began with t ...
, many units for measuring volume were made, such as the ,
amber Amber is fossilized tree resin that has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects."Amber" (2004). In Ma ...
, coomb, and seam. The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by
Henry III of England Henry III (1 October 1207 – 16 November 1272), also known as Henry of Winchester, was King of England, Lord of Ireland, and Duke of Aquitaine from 1216 until his death in 1272. The son of John, King of England, King John and Isabella of ...
. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the '' London Pharmacopoeia'' (medicine compound catalog) adopted the Roman gallon or '' congius'' as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between . Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate the volume of any object. He devised the
Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
, which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat,
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament of the United Kingdom, ...
,
Isaac Barrow Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem o ...
, James Gregory,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
,
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
and Maria Gaetana Agnesi in the 17th and 18th centuries to form the modern integral calculus that is still being used in the 21st century.

## Metrication and redefinitions

On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the '' stère'' (1 m3) for volume of firewood; the '' litre'' (1 dm3) for volumes of liquid; and the ''
gram The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram. Originally defined as of 1795 as "the absolute weight of a volume Volume is a measure ...
me'', for mass—defined as the mass of one cubic centimetre of water at maximum density, at about . Thirty years later in 1824, the
imperial gallon The gallon is a unit of measurement, unit of volume in imperial units and United States customary units. Three different versions are in current use: *the imperial gallon (imp gal), defined as , which is or was used in the United Kingdom, Ire ...
was defined to be the volume occupied by ten pounds of water at . This definition was further refined until the United Kingdom's Weights and Measures Act 1985, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded the metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. The definition of the metre was redefined again in 1983 to use the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
and
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
(which is derived from the
caesium standard The caesium standard is a primary frequency standard in which the Absorption (electromagnetic radiation), photon absorption by transitions between the two hyperfine level, hyperfine ground states of caesium-133 atoms is used to control the output ...
) and reworded for clarity in 2019.

# Measurement

The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as
gourd Gourds include the fruits of some flowering plant species in the family Cucurbitaceae, particularly ''Cucurbita'' and ''Lagenaria''. The term refers to a number of species and subspecies, many with hard shells, and some without. One of the earli ...
s, sheep or pig
stomach The stomach is a muscular, Organ (anatomy), hollow organ in the gastrointestinal tract of humans and many other animals, including several invertebrates. The stomach has a dilated structure and functions as a vital organ in the Digestion, dige ...
s, and bladders. Later on, as
metallurgy Metallurgy is a domain of Materials science, materials science and engineering that studies the physical and chemical behavior of metallic Chemical element, elements, their Inter-metallic alloy, inter-metallic compounds, and their mixtures, which ...
and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method is common for measuring small volume of fluids or
granular material A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compos ...
s, by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients. Air displacement pipette is used in
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of Cell (biology), cells that proce ...
and
biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology a ...
to measure volume of fluids at the microscopic scale. Calibrated
measuring cup A measuring cup is a List of food preparation utensils, kitchen utensil used primarily to measure the volume of liquid or bulk solid cooking ingredients such as flour and sugar, especially for volumes from about 50 millilitre, mL (2 flui ...
s and
spoons Spoons may refer to: * Spoon, a utensil commonly used with soup * Spoons (card game), the card game of Donkey, but using spoons Film and TV *Spoons (TV series), ''Spoons'' (TV series), a 2005 UK comedy sketch show *Spoons, a minor character fro ...
are adequate for cooking and daily life applications, however, they are not precise enough for laboratories. There, volume of liquids is measured using
graduated cylinder A graduated cylinder, also known as a measuring cylinder or mixing cylinder, is a common piece of laboratory equipment used to measure the volume of a liquid. It has a narrow cylindrical shape. Each marked line on the graduated cylinder represent ...
s,
pipette A pipette (sometimes spelled as pipett) is a laboratory tool commonly used in chemistry, biology and medicine to transport a measured volume of liquid, often as a media dispenser. Pipettes come in several designs for various purposes with diffe ...
s and volumetric flasks. The largest of such calibrated containers are petroleum
storage tank Storage tanks are containers that hold liquids, compressed gases (gas tank; or in U.S.A "pressure vessel", which is not typically labeled or regulated as a storage tank) or mediums used for the short- or long-term storage of heat or cold. The t ...
s, some can hold up to of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in a
reservoir A reservoir (; from French language, French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to water storage, store fresh water or it may be a natural formation. Reservoirs can be created in a num ...
, the container's volume is modeled by shapes and calculated using mathematics. 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 mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
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 membership oracle.

## Units

The general form of a unit of volume is the
cube In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
(''x''3) of a unit of . For instance, if the
metre The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
(m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m3). Thus, volume is a SI derived unit and its unit dimension is L3. The metric units of volume uses
metric prefix A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
es, strictly in powers of ten. When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm3 = 2.3 (cm)3 = 2.3 (0.01 m)3 = 0.0000023 m3 (five zeros). Commonly used prefixes for cubed length units are the cubic millimetre (mm3), cubic centimetre (cm3), cubic decimetre (dm3), cubic metre (m3) and the cubic kilometre (km3). The conversion between the prefix units are as follows: 1000 mm3 = 1 cm3, 1000 cm3 = 1 dm3, and 1000 dm3 = 1 m3. The
metric system The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
also includes the litre (L) as a unit of volume, where 1 L = 1 dm3 = 1000 cm3 = 0.001 m3. For the litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Litres are most commonly used for items (such as fluids and solids that can be poured) which are measured by the capacity or size of their container, whereas cubic metres (and derived units) are most commonly used for items measured either by their dimensions or their displacements. Various other imperial or U.S. customary units of volume are also in use, including: * cubic inch, cubic foot, cubic yard, acre-foot, cubic mile; * minim, drachm,
fluid ounce A fluid ounce (abbreviated fl oz, fl. oz. or oz. fl., old forms ℥, fl ℥, f℥, ƒ ℥) is a unit of volume (also called ''capacity'') typically used for measuring Liquid, liquids. The imperial units, British Imperial, the United States custo ...
,
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 ...
; *
teaspoon A teaspoon (tsp.) is an item of cutlery. It is a small spoon that can be used to stir a cup of tea or coffee, or as a tool for Cooking measures, measuring volume. The size of teaspoons ranges from about . For cooking purposes and dosing of med ...
,
tablespoon A tablespoon (tbsp. , Tbsp. , Tb. , or T.) is a large spoon. In many English-speaking regions, the term now refers to a large spoon used for serving; however, in some regions, it is the largest type of spoon used for eating. By extension, the ter ...
; *
gill A gill () is a respiration organ, respiratory organ that many aquatic ecosystem, aquatic organisms use to extract dissolved oxygen from water and to excrete carbon dioxide. The gills of some species, such as hermit crabs, have adapted to allow r ...
, quart, gallon,
barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wood Wood is a porous and fibrous structural tissue found in the Plant stem, stems and roots of trees a ...
; * cord,
peck A peck is an Imperial unit, imperial and United States customary units, United States customary unit of dry measure, dry volume, equivalent to 2 dry gallons or 8 dry quarts or 16 dry pints. An imperial peck is equivalent to 9.09 liters and a US ...
, bushel, hogshead. The smallest volume known to be occupied by matter is probably the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
, with its radius is known to be smaller than 1  femtometre. This means its volume must be smaller than , though the exact value is still under debate as of 2019 as the proton radius puzzle. The van der Waals volume of a
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
atom is far larger, which ranges from to as a sphere with a radius between 100 and 120
picometre The picometre (international American and British English spelling differences#-re, -er, spelling as used by the International Bureau of Weights and Measures; SI symbol: pm) or picometer (American and British English spelling differences#-re, - ...
s. At the other end of the scale, the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
has a volume of around . The largest possible volume in the
observable universe The observable universe is a Ball (mathematics), ball-shaped region of the universe comprising all matter that can be observation, observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electroma ...
is the observable universe itself, at by a sphere of in radius.

## Capacity and volume

Capacity is the maximum amount of material that a container can hold, measured in volume or
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
. However, the contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a tank that can hold of
fuel oil Fuel oil is any of various fractional distillation, fractions obtained from the distillation of petroleum (crude oil). Such oils include distillates (the lighter fractions) and residue (chemistry), residues (the heavier fractions). Fuel oils inclu ...
will not be able to contain the same of
naphtha Naphtha ( or ) is a flammable liquid hydrocarbon mixture. Mixtures labelled ''naphtha'' have been produced from natural gas condensates, petroleum distillates, and the distillation of coal tar and peat. In different industries and regions ''n ...
, due to naphtha's lower density and thus larger volume.

# Calculation

## Basic shapes

This is a list of volume formulas of basic shapes: * Cone$\frac\pi r^3$, where $r$ is the base's radius *
Cube In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
$a^3$, where $a$ is the side's length; *
Cuboid 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 ...
$abc$, where $a$, $b$, and $c$ are the sides' length; *
Cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder ...
$\pi r^2 h$, where $r$ is the base's radius and $h$ is the cone's height; *
Ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
$\frac\pi abc$, where $a$, $b$, and $c$ are the semi-major and semi-minor axes' length; *
Sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
$\frac\pi r^3$, where $r$ is the radius; *
Parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean ...
$abc\sqrt$, where $a$, $b$, and $c$ are the sides' length,$K = 1 + 2\cos(\alpha)\cos(\beta)\cos(\gamma) - \cos^2(\alpha) - \cos^2(\beta) - \cos^2(\gamma)$, and $\alpha$, $\beta$, and $\gamma$ are angles between the two sides; * Prism$Bh$, where $B$ is the base's area and $h$ is the prism's height; *
Pyramid A pyramid (from el, πυραμίς ') is a Nonbuilding structure, structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a Pyramid (geometry), pyramid in the geometric sense. The base o ...
$\fracBh$, where $B$ is the base's area and $h$ is the pyramid's height; *
Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...
$a^3$, where $a$ is the side's length.

## Integral calculus

The calculation of volume is a vital part of
integral 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 ...
calculus. One of which is calculating the volume of solids of revolution, by rotating a
plane curve In mathematics, a plane curve is a curve in a plane (geometry), plane that may be either a Plane (mathematics), Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise ...
around a
line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...
on the same plane. The washer or disc integration method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:$V = \pi \int_a^b \left, f(x)^2 - g(x)^2\\,dx$where $f(x)$ and $g(x)$ are the plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:$V = 2\pi \int_a^b x , f(x) - g(x), \, dx$ The volume of a
region In geography Geography (from Ancient Greek, Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lan ...
''D'' in three-dimensional space is given by the triple or volume integral of the constant function $f\left(x,y,z\right) = 1$ over the region. It is usually written as: $\iiint_D 1 \,dx\,dy\,dz.$ In cylindrical coordinates, the volume integral is $\iiint_D r\,dr\,d\theta\,dz,$ In
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
(using the convention for angles with $\theta$ as the azimuth and $\varphi$ measured from the polar axis; see more on conventions), the volume integral is $\iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .$

## Geometric modeling

A
polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangle A triangle is a polygon with three Edge (geometry), edges a ...
is a representation of the object's surface, using
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 tw ...
s. The volume mesh explicitly define its volume and surface properties.

## 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. It uses the techniques of differential calculus, integral calculus, linear algebra ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a volume form on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topolog ...
is a
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
of top degree (i.e., whose degree is equal to the dimension of the
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclo ...
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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
. Integrating the volume form gives the volume of the manifold according to that form. An
oriented In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
pseudo-Riemannian manifold 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. It uses the techniques of differential calculus, inte ...
has a natural volume form. In local coordinates, it can be expressed as $\omega = \sqrt \, dx^1 \wedge \dots \wedge dx^n ,$ where the $dx^i$ are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold, and $g$ is the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In p ...
of the matrix representation of the metric tensor on the manifold in terms of the same basis.

# Derived quantities

*
Density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
is the substance's
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
per unit volume, or total mass divided by total volume. * Specific volume is total volume divided by mass, or the inverse of density. * The volumetric flow rate or discharge is the volume of fluid which passes through a given surface per unit time. * The
volumetric heat capacity The volumetric heat capacity of a material is the heat capacity of a sample of the substance divided by the volume of the sample. It is the amount of energy that must be added, in the form of heat, to one unit of volume of the material in order ...
is the
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
of the substance divided by its volume.