Volume is a measure of occupied
three-dimensional space.
It is often quantified numerically using
SI derived units (such as the
cubic metre and
litre) or by various
imperial
Imperial is that which relates to an empire, emperor, or imperialism.
Imperial or The Imperial may also refer to:
Places
United States
* Imperial, California
* Imperial, Missouri
* Imperial, Nebraska
* Imperial, Pennsylvania
* Imperial, Texas
...
or
US customary units (such as the
gallon,
quart,
cubic inch). The definition of
length (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 (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 inform ...
shapes can have its volume easily calculated using
arithmetic 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 objects have no volume; in
fourth and higher dimensions, an analogous concept to the normal volume is the hypervolume.
History
Ancient history
The precision of volume measurements in the ancient period usually ranges between .
The earliest evidence of volume calculation came from
ancient Egypt and
Mesopotamia as mathematical problems, approximating volume of simple shapes such as
cuboids,
cylinders,
frustum and
cones. These math problems have been written in the
Moscow Mathematical Papyrus (c. 1820 BCE).
In the
Reisner Papyrus, ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material.
The Egyptians use their units of length (the
cubit,
palm,
digit) 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,
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.
A century later,
Archimedes () devised approximate volume formula of several shapes used the
method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by
Liu Hui in the 3rd century CE,
Zu Chongzhi in the 5th century CE, the
Middle East and
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
.
Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object.
Though highly popularized, Archimedes probably does not submerge the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved.
Instead, he likely have devised a primitive form of a
hydrostatic balance. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a
weighing scale submerged underwater, which will tip accordingly due to the
Archimedes' principle.
Calculus and standardization of units
In the
Middle Ages, many units for measuring volume were made, such as the
sester,
amber,
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. 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 applied the philosophy of modern integral calculus to calculate the volume of any object. He devised the
Cavalieri's principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:
* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
, which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by
Pierre de Fermat,
John Wallis,
Isaac Barrow,
James Gregory,
Isaac Newton,
Gottfried Wilhelm Leibniz and
Maria Gaetana Agnesi in the 17th and 18th centuries to form the modern integral calculus that is still being used in the 21st century.
Metrication and redefinitions
On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the ''
stère'' (1 m
3) for volume of firewood; the ''
litre'' (1 dm
3) for volumes of liquid; and the ''
gramme'', for mass—defined as the mass of one cubic centimetre of water at maximum density, at about . Thirty years later in 1824, the
imperial gallon was defined to be the volume occupied by ten
pounds of water at .
This definition was further refined until the United Kingdom's
Weights and Measures Act 1985, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water.
The 1960 redefinition of the metre from the
International Prototype Metre to the orange-red
emission line of
krypton-86 atoms unbounded the metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre.
The definition of the metre was redefined again in 1983 to use the
speed of light and
second (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
containers found in nature, such as
gourds, sheep or pig
stomachs, and
bladders. Later on, as
metallurgy and
glass production improved, small volumes nowadays are usually measured using standardized human-made containers.
This method is common for measuring small volume of fluids or
granular materials, by using a
multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure
cooking ingredient
Cooking, cookery, or culinary arts is the art, science and craft of using heat to prepare food for consumption. Cooking techniques and ingredients vary widely, from grilling food over an open fire to using electric stoves, to baking in variou ...
s.
Air displacement pipette is used in
biology 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. There, volume of liquids is measured using
graduated cylinders,
pipettes 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, 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 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.
Units
The general form of a unit of volume is the
cube (''x''
3) 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 (m
3). Thus, volume is a
SI derived unit and its
unit dimension is L
3. The metric units of volume uses
metric prefixes, strictly in
powers of ten
A power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The fi ...
. 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 cm
3 = 2.3 (cm)
3 = 2.3 (0.01 m)
3 = 0.0000023 m
3 (five zeros).
Commonly used prefixes for cubed length units are the cubic millimetre (mm
3), cubic centimetre (cm
3), cubic decimetre (dm
3), cubic metre (m
3) and the cubic kilometre (km
3). The conversion between the prefix units are as follows: 1000 mm
3 = 1 cm
3, 1000 cm
3 = 1 dm
3, and 1000 dm
3 = 1 m
3.
The
metric system also includes the
litre (L) as a unit of volume, where 1 L = 1 dm
3 = 1000 cm
3 = 0.001 m
3.
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
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.
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, Texas
...
or
U.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.
The smallest volume known to be occupied by matter is probably the
proton, with its radius is known to be smaller than 1
femtometre. 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 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 has a volume of around .
The largest possible volume in the
observable universe is the observable universe itself, at by a sphere of in radius.
Capacity and volume
Capacity is the maximum amount of material that a container can hold, measured in volume or
weight. However, the contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a tank that can hold of
fuel oil 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 –
, where
is the
base's radius
*
Cube –
, where
is the side's length;
*
Cuboid –
, where
,
, and
are the sides' length;
*
Cylinder –
, where
is the base's radius and
is the cone's height;
*
Ellipsoid –
, where
,
, and
are the
semi-major and semi-minor axes' length;
*
Sphere –
, where
is the radius;
*
Parallelepiped –
, where
,
, and
are the sides' length,
, and
,
, and
are angles between the two sides;
*
Prism –
, where
is the base's area and
is the prism's height;
*
Pyramid –
, where
is the base's area and
is the pyramid's height;
*
Tetrahedron –
, where
is the side's length.
Integral calculus
The calculation of volume is a vital part of
integral calculus. One of which is calculating the volume of
solids of revolution, by rotating a
plane curve around a
line 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:
where
and
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:
The volume of a
region ''D'' in
three-dimensional space is given by the triple or
volume integral of the constant
function over the region. It is usually written as:
In
cylindrical coordinates, the
volume integral is
In
spherical coordinates (using the convention for angles with
as the azimuth and
measured from the polar axis; see more on
conventions), the volume integral is
Geometric modeling
A
polygon mesh is a representation of the object's surface, using
polygons. The
volume mesh
In 3D computer graphics and modeling, volumetric meshes are a polygonal representation of the interior volume of an object. Unlike polygon meshes, which represent only the surface as polygons, volumetric meshes also discretize the interior struct ...
explicitly define its volume and surface properties.
Differential geometry
In
differential geometry, 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
where the
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 ...
s that form a positively oriented basis for the
cotangent bundle of the manifold, and
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
Discharge may refer to
Expel or let go
* Discharge, the act of firing a gun
* Discharge, or termination of employment, the end of an employee's duration with an employer
* Military discharge, the release of a member of the armed forces from ser ...
is the volume of fluid which passes through a given surface per unit time.
* The
volumetric heat capacity is the
heat capacity of the substance divided by its volume.
See also
*
Baggage allowance
*
Banach–Tarski paradox
*
Dimensional weight
Dimensional weight, also known as volumetric weight, is a pricing technique for commercial freight transport (including courier and postal services), which uses an estimated weight that is calculated from the length, width and height of a packag ...
*
Dimensioning
Dimensioning is the process of measuring either the area or the volume that an object occupies. It is the method of calculating capacity for the storage, handling, transporting and invoicing of goods. Vehicles and storage units have both volume and ...
Notes
References
External links
*
*
{{Authority control