Volume is ameasure ofregions inthree-dimensional space.^[1] It is often quantified numerically usingSI derived units (such as thecubic metre andlitre) or by variousimperial orUS customary units (such as thegallon,quart,cubic inch). The definition oflength and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount offluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Bymetonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g.,bounding volume).^[2][3]

Volume
Ameasuring cup can be used to measure volumes ofliquids. This cup measures volume in units ofcups,fluid ounces, andmillilitres.
Common symbols	V
SI unit	cubic metre
Other units	Litre,fluid ounce,gallon,quart,pint,tsp,fluid dram,in3,yd3,barrel
InSI base units	m3
Extensive?	yes
Intensive?	no
Conserved?	yes forsolids andliquids, no forgases, andplasma[a]
Behaviour under coord transformation	conserved
Dimension	L3

In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simplethree-dimensional shapes can have their volume easily calculated usingarithmeticformulas. Volumes of more complicated shapes can be calculated withintegral calculus if a formula exists for the shape's boundary.Zero-,one- andtwo-dimensional objects have no volume; infour and higher dimensions, an analogous concept to the normal volume is the hypervolume.

The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz).^[4]: 8 The earliest evidence of volume calculation came fromancient Egypt andMesopotamia as mathematical problems, approximating volume of simple shapes such ascuboids,cylinders,frustum andcones. These math problems have been written in theMoscow Mathematical Papyrus (c. 1820 BCE). In theReisner 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.^[4]: 116 The Egyptians use their units of length (thecubit,palm,digit) to devise their units of volume, such as the volume cubit^[4]: 117 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).^[4]: 117

The last three books ofEuclid'sElements, written in around 300 BCE, detailed the exact formulas for calculating the volume ofparallelepipeds, cones,pyramids, cylinders, andspheres. The formula were determined by prior mathematicians by using a primitive form ofintegration, by breaking the shapes into smaller and simpler pieces. A century later,Archimedes (c. 287 – 212 BCE) devised approximate volume formula of several shapes using themethod of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently byLiu Hui in the 3rd century CE,Zu Chongzhi in the 5th century CE, theMiddle East andIndia.

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.^[5] Instead, he likely have devised a primitive form of ahydrostatic balance. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of aweighing scale submerged underwater, which will tip accordingly due to theArchimedes' principle.^[6]

In theMiddle Ages, many units for measuring volume were made, such as thesester,amber,coomb, andseam. The sheer quantity of such units motivated British kings to standardize them, culminated in theAssize of Bread and Ale statute in 1258 byHenry III of England. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel.^{[4]: 73–74} In 1618, theLondon Pharmacopoeia (medicine compound catalog) adopted the Roman gallon^[7] orcongius^[8] as a basic unit of volume and gave a conversion table to the apothecaries' units of weight.^[7] Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz).^[4]: 8

Around the early 17th century,Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate the volume of any object. He devisedCavalieri'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 byPierre de Fermat,John Wallis,Isaac Barrow,James Gregory,Isaac Newton,Gottfried Wilhelm Leibniz andMaria Gaetana Agnesi in the 17th and 18th centuries to form the modern integral calculus, which remains in use in the 21st century.

On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: thestère (1 m³) for volume of firewood; thelitre (1 dm³) for volumes of liquid; and thegramme, for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice.^[9] Thirty years later in 1824, theimperial gallon was defined to be the volume occupied by tenpounds of water at 17 °C (62 °F). This definition was further refined until the United Kingdom'sWeights and Measures Act 1985, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water.^[10]

The 1960 redefinition of the metre from theInternational Prototype Metre to the orange-redemission line ofkrypton-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.^[11] The definition of the metre was redefined again in 1983 to use thespeed of light andsecond (which is derived from thecaesium standard) andreworded for clarity in 2019.^[12]

As ameasure of theEuclidean three-dimensional space, volume cannot be physically measured as a negative value, similar tolength andarea. Like all continuousmonotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral toCavalieri's principle and to theinfinitesimal calculus of three-dimensional bodies.^[13] A 'unit' of infinitesimally small volume in integral calculus is thevolume element; this formulation is useful when working with differentcoordinate systems, spaces andmanifolds.

The oldest way to roughly measure a volume of an object is using the human body, such as using hand size andpinches. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durablecontainers found in nature, such asgourds, sheep or pigstomachs, andbladders. Later on, asmetallurgy andglass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method is common for measuring small volume of fluids orgranular materials, by using amultiple 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 measurecooking ingredients.

Air displacement pipette is used inbiology andbiochemistry to measure volume of fluids at the microscopic scale.^[14] Calibratedmeasuring cups andspoons are adequate for cooking and daily life applications, however, they are not precise enough forlaboratories. There, volume of liquids is measured usinggraduated cylinders,pipettes andvolumetric flasks. The largest of such calibrated containers are petroleumstorage tanks, some can hold up to 1,000,000 bbl (160,000,000 L) 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 areservoir, the container's volume is modeled by shapes and calculated using mathematics.

To ease calculations, a unit of volume is equal to the volume occupied by aunit cube (with a side length of one). Because the volume occupies three dimensions, if themetre (m) is chosen as a unit of length, the corresponding unit of volume is thecubic metre (m³). The cubic metre is also aSI derived unit.^[15] Therefore, volume has aunit dimension of L³.^[16]

The metric units of volume usesmetric prefixes, strictly inpowers 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).^[17]: 143

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.^[1] Themetric system also includes thelitre (L) as a unit of volume, where 1 L = 1 dm³ = 1000 cm³ = 0.001 m³.^[17]: 145 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.^[1]

Various otherimperial orU.S. customary units of volume are also in use, including:

• cubic inch,cubic foot,cubic yard,acre-foot,cubic mile;
• minim,drachm,fluid ounce,pint;
• teaspoon,tablespoon;
• gill,quart,gallon,barrel;
• cord,peck,bushel,hogshead.

Capacity is the maximum amount of material that a container can hold, measured in volume orweight. 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 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) offuel oil will not be able to contain the same 7,200 t (15,900,000 lb) ofnaphtha, due to naphtha's lower density and thus larger volume.^{[18]: 390–391}

For many shapes such as thecube,cuboid andcylinder, they have an essentially the same volume calculation formula as one for theprism: thebase of the shape multiplied by itsheight.

The calculation of volume is a vital part ofintegral calculus. One of which is calculating the volume ofsolids of revolution, by rotating aplane curve around aline on the same plane. The washer ordisc integration method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:$${\displaystyle V=\pi {\int }_a^b\left|f(x{)}^2-g(x{)}^2\right|dx}$$ where$f(x)$  and$g(x)$  are the plane curve boundaries.^{[19]: 1, 3} Theshell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:^[19]: 6$${\displaystyle V=2\pi {\int }_a^{b}x|f(x)-g(x)|dx}$$  The volume of aregionD inthree-dimensional space is given by the triple orvolume integral of the constantfunction${\displaystyle f(x,y,z)=1}$  over the region. It is usually written as:^{[20]: Section 14.4}$${\displaystyle {\iiint }_{D}1dxdydz.}$$ 

Incylindrical coordinates, thevolume integral is$${\displaystyle {\iiint }_{D}rdrd\theta dz,}$$ 

Inspherical coordinates (using the convention for angles with${\displaystyle \theta }$  as the azimuth and${\displaystyle \phi }$  measured from the polar axis; see more onconventions), the volume integral is$${\displaystyle {\iiint }_D{\rho }^2\sin \phi d\rho d\theta d\phi .}$$ 

Apolygon mesh is a representation of the object's surface, usingpolygons. Thevolume mesh explicitly define its volume and surface properties.

• Density is the substance'smass per unit volume, or total mass divided by total volume.^[21]
• Specific volume is total volume divided by mass, or the inverse of density.^[22]
• Thevolumetric flow rate ordischarge is the volume of fluid which passes through a given surface per unit time.
• Thevolumetric heat capacity is theheat capacity of the substance divided by its volume.

• Banach–Tarski paradox
• Dimensional weight
• Dimensioning

•  Perimeters, Areas, Volumes at Wikibooks
•  Volume at Wikibooks