Volume

# Volume

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Volume is the quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

of three-dimensional space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

enclosed by some closed boundary, for example, the space that a substance (solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...

, liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

, gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

, or plasma
Plasma
Plasma may refer to:* Blood plasma, the yellow-colored liquid component of blood, in which blood cells are suspended* Plasma , an ionized state of matter similar to a gas...

) or shape occupies or contains.
Volume is often quantified numerically using the SI derived unit
SI derived unit
The International System of Units specifies a set of seven base units from which all other units of measurement are formed, by products of the powers of base units. These other units are called SI derived units, for example, the SI derived unit of area is square metre , and of density is...

, the cubic metre
Cubic metre
The cubic metre is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère...

. The volume of a container
Container
Container may refer to:* Items used to contain, store, and transport products, such as:** Carton** Bottle** Can , several meanings* Shipping containers include** Crate** Wooden box...

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 formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

s. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

) and two-dimensional shapes (such as squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

) 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 (fluid)
In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place. The volume of the fluid displaced can then be measured, as in the illustration, and from this the volume of the immersed object can be deduced .An object that sinks...

. 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 one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive
Additive function
In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...

.

In differential geometry, volume is expressed by means of the volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

, and is an important global Riemannian
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

.
In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, volume is a fundamental parameter
Gas volume
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...

, and is a conjugate variable
Conjugate variables (thermodynamics)
In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials are expressed in terms of conjugate pairs....

to pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

.

## Units

Any unit of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

gives a corresponding unit of volume, namely the volume of a cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

whose side has the given length. For example, a cubic centimetre
Cubic centimetre
A cubic centimetre is a commonly used unit of volume extending the derived SI-unit cubic metre, and corresponds to the volume of a cube measuring 1 cm × 1 cm × 1 cm...

(cm3) would be the volume of a cube whose sides are one centimetre
Centimetre
A centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively...

(1 cm) in length.

In the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI), the standard unit of volume is the cubic metre (m3). The metric system
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...

also includes the litre
Litre
pic|200px|right|thumb|One litre is equivalent to this cubeEach side is 10 cm1 litre water = 1 kilogram water The litre is a metric system unit of volume equal to 1 cubic decimetre , to 1,000 cubic centimetres , and to 1/1,000 cubic metre...

(L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,

so
1 cubic metre = 1000 litres.

Small amounts of liquid are often measured in millilitres, where
1 millilitre = 0.001 litres = 1 cubic centimetre.

Various other traditional units of volume are also in use, including the cubic inch
Cubic inch
The cubic inch is a unit of measurement for volume in the Imperial units and United States customary units systems. It is the volume of a cube with each of its 3 sides being one inch long....

, the cubic foot
Cubic foot
The cubic foot is an Imperial and US customary unit of volume, used in the United States and the United Kingdom. It is defined as the volume of a cube with sides of one foot in length.-Conversions:- Symbols :...

, the cubic mile
Cubic mile
A cubic mile is an imperial / U.S. customary 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 in length....

, the teaspoon
Teaspoon
A teaspoon, an item of cutlery, is a small spoon, commonly part of a silverware place setting, suitable for stirring and sipping the contents of a cup of tea or coffee...

, the tablespoon
Tablespoon
A tablespoon is a type of large spoon usually used for serving. A tablespoonful, the capacity of one tablespoon, is commonly used as a measure of volume in cooking...

, the fluid ounce
Fluid ounce
A fluid ounce is a unit of volume equal to about 28.4 mL in the imperial system or about 29.6 mL in the US system. The fluid ounce is distinct from the ounce, which measures mass...

, the fluid dram, the gill, the pint
Pint
The pint is a unit of volume or capacity that was once used across much of Europe with values varying from state to state from less than half a litre to over one litre. Within continental Europe, the pint was replaced with the metric system during the nineteenth century...

, the quart
Quart
The quart is a unit of volume equal to a quarter of a gallon, two pints, or four cups. Since gallons of various sizes have historically been in use, quarts of various sizes have also existed; see gallon for further discussion. Three of these kinds of quarts remain in current use, all approximately...

, the gallon
Gallon
The gallon is a measure of volume. Historically it has had many different definitions, but there are three definitions in current use: the imperial gallon which is used in the United Kingdom and semi-officially within Canada, the United States liquid gallon and the lesser used United States dry...

, the minim
Minim (unit)
The minim is a unit of volume in both the imperial and US customary systems of measurement. Specifically it is of a fluidram or of a fluid ounce....

, the barrel
Barrel (unit)
A barrel is one of several units of volume, with dry barrels, fluid barrels , oil barrel, etc...

, the cord, the peck
Peck
A peck is an imperial and U.S. customary unit of dry volume, equivalent to 2 gallons or 8 dry quarts or 16 dry pints. Two pecks make a kenning , and four pecks make a bushel....

, the bushel
Bushel
A bushel is an imperial and U.S. customary unit of dry volume, equivalent in each of these systems to 4 pecks or 8 gallons. It is used for volumes of dry commodities , most often in agriculture...

, and the hogshead
Hogshead
A hogshead is a large cask of liquid . More specifically, it refers to a specified volume, measured in either Imperial units or U.S. customary units, primarily applied to alcoholic beverages such as wine, ale, or cider....

.

## Related terms

Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litre
Litre
pic|200px|right|thumb|One litre is equivalent to this cubeEach side is 10 cm1 litre water = 1 kilogram water The litre is a metric system unit of volume equal to 1 cubic decimetre , to 1,000 cubic centimetres , and to 1/1,000 cubic metre...

s or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units).

Volume and capacity are also distinguished in capacity management
Capacity management
Capacity Management is a process used to manage information technology . Its primary goal is to ensure that IT capacity meets current and future business requirements in a cost-effective manner. One common interpretation of Capacity Management is described in the ITIL framework...

, where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity.

The density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of an object is defined as mass per unit volume. The inverse of density is specific volume
Specific volume
In thermodynamics, the specific volume of a substance is the ratio of the substance's volume to its mass. It is the reciprocal of density:In thermodynamics, the specific volume of a substance is the ratio of the substance's volume to its mass...

which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

where the volume of a working fluid is often an important parameter of a system being studied.

The volumetric flow rate
Volumetric flow rate
The volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...

in fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1]).

## Volume formulas

 Shape Volume formula Variables Cube a = length of any side (or edge) CylinderCylinder (geometry)A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder... r = radius of circular face, h = height PrismPrism (geometry)In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a... B = area of the base, h = height Rectangular prism l = length, w = width, h = height SphereSphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point... r = radius of spherewhich is the integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus... of the surface areaSurface areaSurface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces... of a sphere Ellipsoid a, b, c = semi-axes of ellipsoid PyramidPyramid (geometry)In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base.... B = area of the base, h = height of pyramid ConeCone (geometry)A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base... r = radius of circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.... at base, h = distance from base to tip TetrahedronTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... edge length ParallelepipedParallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts... a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges Any volumetric sweep(calculus required) h = any dimension of the figure,A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.(This will work for any figure if its cross-sectional area can be determined from h). Any rotated figure (washer method)(calculus required) and are functions expressing the outer and inner radii of the function, respectively. Klein bottleKlein bottleIn mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a... No volume—it has no inside.

### Ratio of volumes of a cone, sphere and cylinder of the same radius and height

The above formulas can be used to show that the volumes of a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

, sphere and cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

of the same radius and height are in the ratio 1 : 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere).

The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

.

### Sphere

The volume of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

is the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of infinitesimal circular slabs of thickness dx.
The calculation for the volume of a sphere with center 0 and radius r is as follows.

The surface area of the circular slab is .

The radius of the circular slabs, defined such that the x-axis cuts perpendicularly through them, is;

or

where y or z can be taken to represent the radius of a slab at a particular x value.

Using y as the slab radius, the volume of the sphere can be calculated as

Now

Combining yields gives

This formula can be derived more quickly using the formula for the sphere's surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

, which is .
The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to

=

### Cone

The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies cones as well. But for an explanation using calculus:

The volume of a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

is the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of infinitesimal circular slabs of thickness dx.
The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.

The radius of each circular slab is r if x = 0 and 0 if x = h, and varying linearly in between—that is,

The surface area of the circular slab is then

The volume of the cone can then be calculated as

and after extraction of the constants:

Integrating gives us

### Pyramidal shapes

For any shape in three dimensions in which a single base can be identified and a single peak exists non-planer to the base, the pyramidal rule applies.

The volume of any pyramidal shape is one-third of the product of the base area and the length of the altitude (the line-segment perpendicular to the plane that the base is in) of the peak above the plane of the base.

## See also

• Orders of magnitude (volume)
Orders of magnitude (volume)
The pages linked in the right-hand column contain lists of volumes that are of the same order of magnitude . Rows in the table represent increasing powers of a thousand. -Notes:...

• Length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

• Perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

• Area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

• Measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

• Mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

• Weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...

• Conversion of units
• Dimensional weight
Dimensional weight
Dimensional weight, used in shipping and freight, is a billing technique which takes into account the XYZ axis dimensions of a package.-Significance:Shipping costs have historically been calculated on the basis of gross weight in kilograms or pounds...

• Dimensioning
Dimensioning
Dimensioning is the process of measuring the cubic space that a package or object occupies. It is the method of calculating dimensional weight for the storage, handling, transporting and invoicing of goods. Vehicles and storage units have both volume and weight capacity limits and can easily become...

• Volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

• Volume (thermodynamics)
• Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...