TheInfoList

OR:

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 of an element (i.e., point). This is the inform ...
. 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 p ...
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 ...
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 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 uni ...
(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, Austra ...
,
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 eq ...
, 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 Inte ...
(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 shea ...
(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 19t ...
formulas. 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 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 ...
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 ...
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 cu ...
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 in ...
s, 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 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, 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,
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, quadrilate ...
s, cylinders, and
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
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,
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 i ...
() 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 be ...
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 The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabia (including the Arabian Peninsula and Bahrain), Asia Minor (Asian part of Turkey except Hatay Province), East Thrace (Europe ...
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 th ...
. 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 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.

## 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 ...
, many units for measuring volume were made, such as the sester,
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 King John and Isabella of Angoulême, Henry ...
. 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, 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 is ...
, John Wallis,
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 great ...
,
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 mat ...
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 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 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 water equal to th ...
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 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 theory of relativity, is the upper limit f ...
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) 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
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 ter ...
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 ...
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 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 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 cells that process hereditary ...
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 cups 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 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 diff ...
s and volumetric flasks. The largest of such calibrated containers are petroleum storage tanks, 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 controll ...
, 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 a ...
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 problems or to perform a computation. Algorithms are used as specifications for performing ...
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 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 ...
(''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 Inte ...
. 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 pr ...
(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 ...
(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 p ...
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. 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 decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Interna ...
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 shea ...
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 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 w ...
, cubic yard,
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; * 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 S ...
,
pint The pint (, ; symbol pt, sometimes abbreviated as ''p'') is a unit of volume or capacity in both the imperial and United States customary measurement systems. In both of those systems it is traditionally one eighth of a gallon. The British imp ...
; * teaspoon,
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,
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 eq ...
,
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, Austra ...
,
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, u ...
; * 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 alco ...
. 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_a ...
. 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 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, ...
atom is far larger, which ranges from to as a sphere with a radius between 100 and 120  picometres. 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-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 o ...
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 vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
. 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), bu ...
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 ...
$a^3$, 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 cu ...
$abc$, where $a$, $b$, and $c$ are the sides' length; *
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 in ...
$\pi r^2 h$, 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
$\frac\pi r^3$, 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 Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimenta ...
$Bh$, where $B$ is the base's area and $h$ is the prism's height; *
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, quadrilate ...
$\fracBh$, 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, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
$a^3$, 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 functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
calculus. One of which is calculating the volume of solids of revolution, by rotating a plane curve around a 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:$V = \pi \int_a^b \left, f(x)^2 - g(x)^2\\,dx$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, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
''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 of an element (i.e., point). This is the inform ...
is given by the triple or
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
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 In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
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 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 two t ...
s. The volume mesh 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 multil ...
, 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 ...
is a differential form 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 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 letter rho), although the Latin letter ''D'' can also be used. Mathematica ...
. Integrating the volume form gives the volume of the manifold according to that form. An 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-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 e ...
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 ma ...
of the manifold, and $g$ is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
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 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 letter rho), although the Latin letter ''D'' can also be used. Mathematica ...
is the substance's
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
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 is the volume of fluid which passes through a given surface per unit time. * The volumetric heat capacity is the
heat capacity Heat capacity or thermal capacity is a 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 SI unit of heat capacity is joule per kelvin (J/K). Heat capacity ...
of the substance divided by its volume.