Volume (mathematics)
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Volume is a measure of occupied
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
. 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 The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). ...
) 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 United States customary units form a system of measurement units commonly used in the United States and U.S. territories since being standardized and adopted in 1832. The United States customary system (USCS or USC) developed from English units ...
(such as the
gallon The gallon is a 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, Ireland, Canada, Austral ...
,
quart The quart (symbol: qt) is an English unit of volume equal to a quarter gallon. Three kinds of quarts are currently used: the liquid quart and dry quart of the US customary system and the of the British imperial system. All are roughly equal ...
,
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 height) being one inch long which is equivalent ...
). 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 Three-dimensional space (also: 3D space, 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 ...
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 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 ...
if a formula exists for the shape's boundary. Zero-, one- and
two-dimensional In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
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 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
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,
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
s,
frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are p ...
and
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
s. These math problems have been written in the
Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
(c. 1820 BCE). In the
Reisner Papyrus The Reisner Papyri date to the reign of Senusret I, who was king of ancient Egypt in the 19th century BCE. The documents were discovered by G.A. Reisner during excavations in 1901–04 in Naga ed-Deir in southern Egypt. A total of four papyrus rol ...
, 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 The cubit is an ancient unit of length based on the distance from the elbow to the tip of the middle finger. It was primarily associated with the Sumerians, Egyptians, and Israelites. The term ''cubit'' is found in the Bible regarding No ...
,
palm Palm most commonly refers to: * Palm of the hand, the central region of the front of the hand * Palm plants, of family Arecaceae **List of Arecaceae genera * Several other plants known as "palm" Palm or Palms may also refer to: Music * Palm (ba ...
, 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
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,
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
s, cylinders, and
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 ...
s. 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 The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...
approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
in the 3rd century CE,
Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3. ...
in the 5th century CE, the
Middle East The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabian Peninsula, Arabia (including the Arabian Peninsula and Bahrain), Anatolia, Asia Minor (Asian part of Turkey except Hatay Pro ...
and
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
. 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 In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a
weighing scale A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a ...
submerged underwater, which will tip accordingly due to the
Archimedes' principle Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimede ...
.


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 period of global history. It began with the fall of the Western Roman Empire a ...
, 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 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 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 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 King John and Isabella of Angoulême, Henry a ...
. 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 In Ancient Roman measurement, ''congius'' (pl. ''congii'', from Greek ''konkhion'', diminutive of ''konkhē'', ''konkhos'', "shellful") was a liquid measure that was about 3.48 litres (0.92 U.S. gallons). It was equal to the larger chous of the A ...
'' 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 Bonaventura Francesco Cavalieri ( la, Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infi ...
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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
,
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 Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
,
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 Maria Gaetana Agnesi ( , , ; 16 May 1718 – 9 January 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics profes ...
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 The stere or stère (st) is a unit of volume in the original metric system equal to one cubic metre. The stere is typically used for measuring large quantities of firewood or other cut wood, while the cubic meter is used for uncut wood. The n ...
'' (1 m3) for volume of firewood; the ''
litre The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). ...
'' (1 dm3) for volumes of liquid; and the ''
gram The gram (originally gramme; SI unit symbol g) is a Physical unit, 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 of pure wate ...
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 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, Ireland, Canada, Austral ...
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 Weights and measures acts are acts of the British Parliament determining the regulation of weights and measures. It also refers to similar royal and parliamentary acts of the Kingdoms of England and Scotland and the medieval Welsh states. T ...
, 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 The history of the metre starts with the Scientific Revolution that is considered to have begun with Nicolaus Copernicus's publication of ''De revolutionibus orbium coelestium'' in 1543. Increasingly accurate measurements were required, and sc ...
to the orange-red
emission line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
of
krypton-86 There are 34 known isotopes of krypton (36Kr) with atomic mass numbers from 69 through 102. Naturally occurring krypton is made of five stable isotopes and one () which is slightly radioactive with an extremely long half-life, plus traces of radio ...
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 theory of relativity, is the upper limit ...
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
container A container is any receptacle or enclosure for holding a product used in storage, packaging, and transportation, including shipping. Things kept inside of a container are protected on several sides by being inside of its structure. The term ...
s 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, 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 digestive system. The stomach i ...
s, 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 Glass production involves two main methods – the float glass process that produces sheet glass, and glassblowing that produces bottles and other containers. It has been done in a variety of ways during the history of glass. Glass container ...
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 compose gra ...
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 Piston-driven air displacement pipettes are a type of micropipette, which are tools to handle volumes of liquid in the microliter scale. They are more commonly used in biology and biochemistry, and less commonly in chemistry; the equipment is sus ...
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 cells that process hereditary i ...
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 and ...
to measure volume of fluids at the microscopic scale. Calibrated
measuring cup A measuring cup is a 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  mL (2  fl oz) upwards. Measuring cups are also used ...
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), a 2005 UK comedy sketch show *Spoons, a minor character from ''The Sopranos'' ...
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 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 flask A volumetric flask (measuring flask or graduated flask) is a piece of laboratory apparatus, a type of laboratory flask, calibrated to contain a precise volume at a certain temperature. Volumetric flasks are used for precise dilutions and preparat ...
s. 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 ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including contro ...
, 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 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 In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
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 The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(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 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 pre ...
es, 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 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 The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). ...
(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 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 height) being one inch long which is equivalent ...
,
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 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 ...
,
acre-foot The acre-foot is a non- SI unit of volume equal to about commonly used in the United States in reference to large-scale water resources, such as reservoirs, aqueducts, canals, sewer flow capacity, irrigation water, and river flows. An acre-f ...
,
cubic mile A cubic mile (abbreviation: cu mi or mi3) is an imperial and US customary (non- SI non-metric) unit of volume, used in the United States, Canada and the United Kingdom. It is defined as the volume of a cube with sides of 1 mile (63360 inche ...
; * 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 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 liquids. The British Imperial, the United States customary, and the United St ...
,
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 respiratory organ that many 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 respiration on land provided they are ...
,
quart The quart (symbol: qt) is an English unit of volume equal to a quarter gallon. Three kinds of quarts are currently used: the liquid quart and dry quart of the US customary system and the of the British imperial system. All are roughly equal ...
,
gallon The gallon is a 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, Ireland, Canada, Austral ...
,
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 Cord or CORD may refer to: People * Alex Cord (1933–2021), American actor and writer * Chris Cord (born 1940), American racing driver * Errett Lobban Cord (1894–1974) American industrialist * Ronnie Cord (1943–1986), Brazilian singer * Co ...
,
peck A peck is an imperial and United States customary unit of 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 customary peck is equivalent to 8.81 liters. Two pecks mak ...
,
bushel A bushel (abbreviation: bsh. or bu.) is an imperial and US customary unit of volume based upon an earlier measure of dry capacity. The old bushel is equal to 2 kennings (obsolete), 4 pecks, or 8 dry gallons, and was used mostly for agricult ...
,
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 proton radius puzzle is an unanswered problem in physics relating to the size of the proton. Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m ...
. 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 Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
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 Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
has a volume of around . The largest possible volume in the
observable universe The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
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 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 (geometry), base's radius * Cube – a^3, where a is the side's length; * Cuboid – 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 – \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 – 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 (geometry), Prism – Bh, where B is the base's area and h is the prism's height; * Pyramid (geometry), Pyramid – \fracBh, where B is the base's area and h is the pyramid's height; * Tetrahedron – a^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 Solid of revolution, solids of revolution, by rotating a plane curve around a Line (geometry), line 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\\,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 (mathematics), region ''D'' in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
is given by the triple or volume integral of the constant function (mathematics), function f(x,y,z) = 1 over the region. It is usually written as: \iiint_D 1 \,dx\,dy\,dz. In cylindrical coordinate system, cylindrical coordinates, the volume integral is \iiint_D r\,dr\,d\theta\,dz, In spherical coordinate system, spherical coordinates (using the convention for angles with \theta as the azimuth and \varphi measured from the polar axis; see more on Spherical coordinate system#Conventions, conventions), 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 explicitly define its volume and surface properties.


Differential geometry

In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form 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 on a manifold, density. Integrating the volume form gives the volume of the manifold according to that form. An orientation (space), oriented 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-forms that form a positively oriented basis for the cotangent bundle of the manifold, and g is the determinant of the matrix representation of the metric tensor 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 or Discharge (hydrology), discharge 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 * Dimensioning


Notes


References


External links

* * {{Authority control Volume,