Volume

Volume
Volume
	A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
Common symbols	V
SI unit	Cubic metre [m3]
Other units	Litre, fluid ounce, gallon, quart, pint, tsp, fluid dram, in3, yd3, barrel
In SI base units	1 m3
Dimension	L3

Volume is a scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface.^[1] For example, the space that a substance (solid, liquid, gas, or plasma) or 3D shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes, can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.^[2]

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

History

The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids, cylinders, frustum and cones. These math problems have been written in the Moscow Mathematical Papyrus (c. 1820 BCE).^[3]^: 403 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.^[4]^: 116 The Egyptians use their units of length (the cubit, palm, digit) to devise their units of volume, such as the volume cubit (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 of Euclid's Elements, written in around 300 BCE, detailed the exact formulas for calculating the volume of parallelepipeds, cones, pyramids, cylinders, and spheres. The formula were determined by prior mathematicians by using a primitive form of integration, by breaking the shapes into smaller and simpler pieces.^[3]^: 403

Though highly popularized, Archimedes probably does not submerge the golden crown and measure the water displacement 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 a hydrostatic balance. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle.^[6]

Units

6 volumetric measures from the mens ponderia in Pompeii, a municipal institution for the control of weights and measures (79 A. D.)

The general form of a unit of volume is the cube (x³) of a unit of length. For instance, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m³).^[7] Thus, the unit dimension of volume is L³.^[8] The cubic metre is also the standard unit of volume in the International System of Units (SI).^[9]^: 139 The metric system also includes the litre (L) as a unit of volume, where 1L = 1 dm³ = 1000 cm³ = 0.001 m³.^[9]^: 145

Litres are most commonly used for items (such as fluids and solids that can be poured) which are measured by the capacity or size of their container, whereas cubic metres (and derived units) are most commonly used for items measured either by their dimensions or their displacements.^{[citation needed]}

Small amounts of liquid are often measured in millilitres, where

1 millilitre = 0.001 litres = 1 cubic centimetre.

In the same way, large amounts can be measured in megalitres, where

1 million litres = 1000 cubic metres = 1 megalitre.

Various other traditional units of volume are also in use, including:

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

Derived quantities

The density of an object is defined as the ratio of the mass to the volume.^[10] The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.

The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example, cubic meters per second [m³ s⁻¹]).

Volumetric space is a 3D region having a shape in addition to capacity or volume.

Calculus

In calculus, a branch of mathematics, the volume of a region D in three-dimensional space is given by a triple integral of the constant function $f(x,y,z)=1$ over the region and is usually written as:

\iiint \limits _{D}1\,dx\,dy\,dz.

In cylindrical coordinates, the volume integral is

\iiint \limits _{D}r\,dr\,d\theta \,dz,

In spherical coordinates (using the convention for angles with $\theta$ as the azimuth and $\varphi$ measured from the polar axis; see more on conventions), the volume integral is

\iiint \limits _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .

Formula of basic shapes

Shape	Volume formula^[3]^{: 405–406}	Variables
Cube	$a^{3}$
Cuboid	$abc$
Prism	$Bh$
Parallelepiped	$abc{\sqrt {K}}$ ${\begin{aligned}K=1&+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}$
Pyramid	${\frac {1}{3}}Bh$
Regular tetrahedron	${{\sqrt {2}} \over 12}a^{3}\,$
Cone	${\frac {1}{3}}\pi r^{2}h$
Cylinder	$\pi r^{2}h$
Sphere	${\frac {4}{3}}\pi r^{3}$
Ellipsoid	${\frac {4}{3}}\pi abc$
Solid torus	$2\pi ^{2}Rr^{2}$
Solid body with continuous area	$\int _{a}^{b}A(x)\mathrm {d} x$	–
Solid of revolution	$\pi \int _{a}^{b}\left\|f(x)^{2}-g(x)^{2}\right\|\,dx\,$

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. 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 {|g|}}\,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.

Thermodynamics

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law.

Computation

The task of numerically computing the volume of objects is studied in the field of computational geometry in computer science, investigating efficient algorithms 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.

References

^ "SI Units - Volume". National Institute of Standards and Technology. April 13, 2022.
^ One litre of sugar (about 970 grams) can dissolve in 0.6 litres of hot water, producing a total volume of less than one litre. "Solubility". Retrieved 2010-05-01. Up to 1800 grams of sucrose can dissolve in a liter of water.
^ ^a ^b ^c Treese, Steven A. (2018). History and Measurement of the Base and Derived Units. Cham, Switzerland: Springer Science+Business Media. ISBN 978-3-319-77577-7. LCCN 2018940415. OCLC 1036766223.
^ ^a ^b Imhausen, Annette (2016). Mathematics in Ancient Egypt: A Contextual History. Princeton University Press. ISBN 978-1-4008-7430-9. OCLC 934433864.
^ Rorres, Chris. "The Golden Crown". Drexel University. Archived from the original on 11 March 2009. Retrieved 24 March 2009.
^ Graf, E. H. (2004). "Just what did Archimedes say about buoyancy?". The Physics Teacher. 42 (5): 296–299. Bibcode:2004PhTea..42..296G. doi:10.1119/1.1737965.
^ "Area and Volume". National Institute of Standards and Technology. February 25, 2022.
^ Lemons, Don S. (16 March 2017). A Student's Guide to Dimensional Analysis. New York: Cambridge University Press. p. 38. ISBN 978-1-107-16115-3. OCLC 959922612.
^ ^a ^b The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, ISBN 978-92-822-2272-0
^ "density". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

External links

Perimeters, Areas, Volumes at Wikibooks
Volume at Wikibooks

[1] "SI Units - Volume". National Institute of Standards and Technology. April 13, 2022.

[2] One litre of sugar (about 970 grams) can dissolve in 0.6 litres of hot water, producing a total volume of less than one litre. "Solubility". Retrieved 2010-05-01. Up to 1800 grams of sucrose can dissolve in a liter of water.

[:2-3] Treese, Steven A. (2018). History and Measurement of the Base and Derived Units. Cham, Switzerland: Springer Science+Business Media. ISBN 978-3-319-77577-7. LCCN 2018940415. OCLC 1036766223.

[:3-4] Imhausen, Annette (2016). Mathematics in Ancient Egypt: A Contextual History. Princeton University Press. ISBN 978-1-4008-7430-9. OCLC 934433864.

[inaccuracy-5] Rorres, Chris. "The Golden Crown". Drexel University. Archived from the original on 11 March 2009. Retrieved 24 March 2009.

[:7-6] Graf, E. H. (2004). "Just what did Archimedes say about buoyancy?". The Physics Teacher. 42 (5): 296–299. Bibcode:2004PhTea..42..296G. doi:10.1119/1.1737965.

[7] "Area and Volume". National Institute of Standards and Technology. February 25, 2022.

[8] Lemons, Don S. (16 March 2017). A Student's Guide to Dimensional Analysis. New York: Cambridge University Press. p. 38. ISBN 978-1-107-16115-3. OCLC 959922612.

[:1-9] The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, ISBN 978-92-822-2272-0

[10] "density". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

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