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Volume is a measure of occupied three-dimensional space. It is often quantified numerically using
SI derived unit SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate po ...
s (such as the
cubic metre The cubic metre (in Commonwealth English and international spelling as used by the International Bureau of Weights and Measures) or cubic meter (in American English) is the unit of volume in the International System of Units (SI). Its symbol is m ...
and litre) or by various
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, Pennsylvania * Imperial, Texa ...
or US customary units (such as the gallon, quart, cubic inch). The definition of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
(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 In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
(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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
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 and Mesopotamia as mathematical problems, approximating volume of simple shapes such as
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 ...
s, cylinders, frustum and cones. 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 Digit may refer to: Mathematics and science * Numerical digit, as used in mathematics or computer science ** Hindu-Arabic numerals, the most common modern representation of numerical digits * Digit (anatomy), the most distal part of a limb, such ...
) 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
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 Euclidea ...
s, cones, pyramids, cylinders, and spheres. The formula were determined by prior mathematicians by using a primitive form of
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
, by breaking the shapes into smaller and simpler pieces. A century later,
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
() 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 Middle East and India. 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 weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle.


Calculus and standardization of units

In the Middle Ages, many units for measuring volume were made, such as the
sester The ancient Roman units of measurement were primarily founded on the Hellenic system, which in turn was influenced by the Egyptian system and the Mesopotamian system. The Roman units were comparatively consistent and well documented. Length T ...
, amber, coomb, and
seam Seam may refer to: Science and technology * Seam (geology), a stratum of coal or mineral that is economically viable; a bed or a distinct layer of vein of rock in other layers of rock * Seam (metallurgy), a metalworking process the joins the ends ...
. The sheer quantity of such units motivated British kings to standardize them, culminated in the
Assize of Bread and Ale The Assize of Bread and Ale ( la, Assisa panis et cervisiae) was a 13th-century law in high medieval England, which regulated the price, weight and quality of the bread and beer manufactured and sold in towns, villages and hamlets. It was the firs ...
statute in 1258 by Henry III of England. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the ''
London Pharmacopoeia A pharmacopoeia, pharmacopeia, or pharmacopoea (from the obsolete typography ''pharmacopœia'', meaning "drug-making"), in its modern technical sense, is a book containing directions for the identification of compound medicines, and published by ...
'' (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 and, later, the royal ...
,
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 ...
, James Gregory, Isaac Newton, Gottfried Wilhelm Leibniz 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 '' gramme'', 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 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 and
second The second (symbol: s) is the unit of 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 to 60 seconds ...
(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 Pinches is the surname of the following people: * Barry Pinches (born 1970), English snooker player *Jennifer Pinches (born 1994), British artistic gymnast * John Pinches (1916–2007), English rower, Royal Engineers officer, medallist and author * ...
. 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 gourds, sheep or pig stomachs, and bladders. Later on, as
metallurgy Metallurgy is a domain of materials science and engineering that studies the physical and chemical behavior of metallic elements, their inter-metallic compounds, and their mixtures, which are known as alloys. Metallurgy encompasses both the sc ...
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 materials, 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 ingredient Cooking, cookery, or culinary arts is the art, science and craft of using heat to prepare food for consumption. Cooking techniques and ingredients vary widely, from grilling food over an open fire to using electric stoves, to baking in vario ...
s. Air displacement pipette is used in biology and biochemistry to measure volume of fluids at the microscopic scale. Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for
laboratories A laboratory (; ; colloquially lab) is a facility that provides controlled conditions in which scientific or technological research, experiments, and measurement may be performed. Laboratory services are provided in a variety of settings: physicia ...
. There, volume of liquids is measured using graduated cylinders,
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, 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 algorithms to perform this computation,
approximately An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
or exactly, for various types of objects. For instance, the
convex volume approximation In the analysis of algorithms, several authors have studied the computation of the volume of high-dimensional convex bodies, a problem that can also be used to model many other problems in combinatorial enumeration. Often these works use a black bo ...
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 solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
(''x''3) of a unit of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
. For instance, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the
cubic metre The cubic metre (in Commonwealth English and international spelling as used by the International Bureau of Weights and Measures) or cubic meter (in American English) is the unit of volume in the International System of Units (SI). Its symbol is m ...
(m3). Thus, volume is a
SI derived unit SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate po ...
and its unit dimension is L3. The metric units of volume uses metric prefixes, strictly in
powers of ten A power of 10 is any of the integer exponentiation, powers of the number 10 (number), ten; in other words, ten multiplication, multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is ...
. 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 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
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s 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 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, Pennsylvania * Imperial, Texa ...
or U.S. customary units of volume are also in use, including: * cubic inch,
cubic foot Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system ...
, cubic yard, acre-foot, cubic mile; * minim,
drachm 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, ...
, fluid ounce, pint; * teaspoon, tablespoon; * gill, 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 wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
; * cord, peck, bushel,
hogshead A hogshead (abbreviated "hhd", plural "hhds") is a large cask of liquid (or, less often, of a food commodity). More specifically, it refers to a specified volume, measured in either imperial or US customary measures, primarily applied to alcoho ...
. 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 The magnitudes_.html" ;"title="Magnitude_(mathematics).html" ;"title="atom.html" ;"title="helium helium_atom_and_perspective_Magnitude_(mathematics)">magnitudes_">Magnitude_(mathematics).html"_;"title="atom.html"_;"title="helium_atom">helium_at ...
. 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 The van der Waals radius, ''r'', of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, ...
of a hydrogen atom is far larger, which ranges from to as a sphere with a radius between 100 and 120 
picometre The picometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: pm) or picometer (American spelling) is a unit of length in the International System of Units (SI), equal to , or one trillionth of ...
s. At the other end of the scale, the Earth has a volume of around . The largest possible volume in the observable universe 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. 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 fractions obtained from the distillation of petroleum (crude oil). Such oils include distillates (the lighter fractions) and residues (the heavier fractions). Fuel oils include heavy fuel oil, marine fuel oil (MFO), bun ...
will not be able to contain the same of naphtha, 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 solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
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\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 scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
\frac\pi abc, where a, b, and c are the semi-major and semi-minor axes' length; * Sphere\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 Euclidea ...
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; * PrismBh, where B is the base's area and h is the prism's height; * Pyramid\fracBh, where B is the base's area and h is the pyramid's height; * Tetrahedrona^3, where a is the side's length.


Integral calculus

The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution, by rotating a plane curve around a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
on the same plane. The washer or
disc integration Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method mod ...
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\\,dxwhere 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 ''D'' in three-dimensional space is given by the triple or volume integral of the constant function f(x,y,z) = 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 (using the convention for angles with \theta as the azimuth and \varphi measured from the polar axis; see more on
conventions Convention may refer to: * Convention (norm), a custom or tradition, a standard of presentation or conduct ** Treaty, an agreement in international law * Convention (meeting), meeting of a (usually large) group of individuals and/or companies in a ...
), the volume integral is \iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .


Geometric modeling

A polygon mesh is a representation of the object's surface, using polygons. The
volume mesh In 3D computer graphics and modeling, volumetric meshes are a polygonal representation of the interior volume of an object. Unlike polygon meshes, which represent only the surface as polygons, volumetric meshes also discretize the interior struct ...
explicitly define its volume and surface properties.


Differential geometry

In
differential geometry Differential geometry is a 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 and multili ...
, 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). One ma ...
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, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
) 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. Integrating the volume form gives the volume of the manifold according to that form. An
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
pseudo-Riemannian manifold 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-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
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 of the matrix representation of the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
on the manifold in terms of the same basis.


Derived quantities

* Density is the substance's mass 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 In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
or
discharge Discharge may refer to Expel or let go * Discharge, the act of firing a gun * Discharge, or termination of employment, the end of an employee's duration with an employer * Military discharge, the release of a member of the armed forces from serv ...
is the volume of fluid which passes through a given surface per unit time. * The volumetric heat capacity is the heat capacity of the substance divided by its volume.


See also

* Baggage allowance * Banach–Tarski paradox *
Dimensional weight Dimensional weight, also known as volumetric weight, is a pricing technique for commercial freight transport (including courier and postal services), which uses an estimated weight that is calculated from the length, width and height of a packag ...
*
Dimensioning Dimensioning is the process of measuring either the area or the volume that an object occupies. It is the method of calculating capacity for the storage, handling, transporting and invoicing of goods. Vehicles and storage units have both volume and ...


Notes


References


External links

* * {{Authority control