Volume

Information about Volume

The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

Volumes of straight-edged and circular shapes are calculated using arithmetic formulae. Volumes of other curved shapes are calculated using integral calculus, by approximating the given body with a large amount of small cubes or concentric cylindrical shells, and adding the individual volumes of those shapes. The volume of irregularly shaped objects can be determined by displacement. If an irregularly shaped object floats on water, you will need a heavier object like a rock or metal and attach it on you floating object. This should cause the object to sink. Then, get the volume of the object. Subtract the volume of the attached heavy object and the original findings.

The generalization of volume to arbitrarily many dimensions is called content. In differential geometry, volume is expressed by means of the volume form.

Volume and Capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres 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 a capacity management setting, where capacity is defined as volume over a specified time period.

Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure.

Conjugate variables of thermodynamics
Pressure	Volume
(Stress)	(Strain)
Temperature	Entropy
Chem. potential	Particle no.

Volume formulae

Common equations for volume:
Shape	Equation	Variables
A cube:		s = length of a side
A rectangular prism:		l = length, w = width, h = height
A cylinder (circular prism):		r = radius of circular face, h = height
Any prism that has a constant cross sectional area along the height**:		A = area of the base, h = height
A sphere:	$\text{[math]}$	r = radius of sphere which is the integral of the Surface Area of a sphere
An ellipsoid:		a, b, c = semi-axes of ellipsoid
A pyramid:		l = length, w = width, h = height
A cone (circular-based pyramid):		r = radius of circle at base, h = distance from base to tip
Any figure (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 this will work for any figure if its cross-sectional area can be determined from h (no matter if the prism is slanted or the cross-sections change shape).

(The units of volume depend on the units of length - if the lengths are in metres, the volume will be in cubic metres, etc)

The volume of a parallelepiped is the absolute value of the scalar triple product of the subtending vectors, or equivalently the absolute value of the determinant of the corresponding matrix.

The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.

Volume measures: USA

U.S. customary units of volume:

U.S. fluid ounce, about 29.6 mL
U.S. liquid pint = 16 fluid ounces, or about 473 mL
U.S. dry pint = 1/64 U.S. bushel, or about 551 mL (used for things such as blueberries)
U.S. liquid quart = 32 fluid ounces, or about 946 mL
U.S. dry quart = 1/32 U.S. bushel, or about 1.101 L
U.S. liquid gallon = 128 fluid ounces or four U.S. quarts, about 3.785 L
U.S. dry gallon = 1/8 U.S. bushel, or about 4.405 L
U.S. (dry level) bushel = 2150.42 cubic inches, or about 35.239 L

The acre foot is often used in measuring the volume of water in a reservoir or an aquifer. It is the volume of water that would cover an area of one acre to a depth of one foot. It is equivalent to 43,560 cubic feet or exactly 1233.481 837 547 52 m³.

cubic inch = 16.387064 cm³
cubic foot = 1,728 in³ ≈ 28.317 dm³
cubic yard = 27 ft³ ≈ 0.7646 m³
cubic mile = 5,451,776,000 yd³ = 3,379,200 acre-feet ≈ 4.168 km³

Volume measures: UK

The UK is undergoing metrication and is increasingly using the SI metric system's units of volume, i.e. cubic meter and litre. However, some former units of volume are still in varying degrees of usage:

Imperial units of volume:

UK fluid ounce, about 28.4 mL (this equals the volume of an avoirdupois ounce of water under certain conditions)
UK pint = 20 fluid ounces, or about 568 mL
UK quart = 40 ounces or two pints1.137 L
UK gallon = 4 quarts, or exactly 4.546 09 L

The quart is now obsolete and the fluid ounce extremely rare. The gallon is only used for transportation uses, (it is illegal for petrol and diesel to be sold by the gallon). The pint is the only Imperial unit that is in everyday use, for the sale of draught beer and cider (bottled and canned beer is mainly sold in SI units) and for milk (this too is increasingly being sold in SI units, mainly Liters).

Volume measures: cooking

Traditional cooking measures for volume also include:

teaspoon = 1/6 U.S. fluid ounce (about 4.929 mL)
teaspoon = 1/6 Imperial fluid ounce (about 4.736 mL)
teaspoon = 5 mL (metric)
tablespoon = ½ U.S. fluid ounce or 3 teaspoons (about 14.79 mL)
tablespoon = ½ Imperial fluid ounce or 3 teaspoons (about 14.21 mL)
tablespoon = 15 mL or 3 teaspoons (metric)
tablespoon = 5 fluidrams (about 17.76 mL) (British)
cup = 8 U.S. fluid ounces or ½ U.S. liquid pint (about 237 mL)
cup = 8 Imperial fluid ounces or ½ fluid pint (about 227 mL)
cup = 250 mL (metric)

Relationship to density

The volume of an object is equal to its mass divided by its average density. This is a rearrangement of the calculation of density as mass per unit volume.

The term specific volume is used for volume divided by mass. This is the reciprocal of the mass density, expressed in units such as cubic meters per kilogram (m³·kg^-1).

External links

FORTRAN code for finding volumes of various shapes

dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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In plane (Euclidean) geometry, a square is circle with four sides.

Classification

A square is a regular quadrilateral. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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cube^[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each . The cube can also be called a regular hexahedron and is one of the five Platonic solids.
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cylinder is a quadric surface, with the following equation in Cartesian coordinates:

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b).
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In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place, so that it can be weighed.

An object that sinks also displaces an amount of fluid equal to the object's volume.
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Content may refer to:

Content (media and publishing), information and experiences created to benefit audiences in contexts that they value
Volume generalized to arbitrarily many dimensions in mathematics and physics

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In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf.
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In mathematics, a volume form is a nowhere zero differential n-form on an oriented n-manifold. Every volume form defines a measure on the manifold, and thus a means to calculate volumes in a generalized sense.
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The litre or liter (see spelling differences) is a unit of volume. There are two official symbols, namely the Latin letter L both in lower and upper case: l and L.
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cubic metre (symbol m³) is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. In the United States it is spelled cubic meter. An alternate name, which allowed a different usage with SI prefixes, was the stère.
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Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on
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conjugate variables such as pressure/volume or temperature/entropy. In fact all thermodynamic potentials are expressed in terms of conjugate pairs.

For a mechanical system, a small increment of energy is the product of a force times a small displacement.
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Pressure (symbol: p) is the force per unit area applied on a surface in a direction perpendicular to that surface.

Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
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Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
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strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain is calculated by first assuming a change between two body states: the beginning state and the final state.
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trillion fold).]]

Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
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Ice melting - a classic example of entropy increasing^[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows:

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The particle number, N, is the number of so called 'elementary particles' (or elementary constituents) in a thermodynamical system. The particle number is a fundamental parameter in thermodynamics and it is conjugate to the chemical potential.
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equation is a mathematical statement, in symbols, that two things are the same (or equivalent). Equations are written with an equal sign, as in

The equation above is an example of an equality: a proposition which states that two constants are equal.
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prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.
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cylinder is a quadric surface, with the following equation in Cartesian coordinates:

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b).
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A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. In mathematics, a sphere is the set of all points in three-dimensional space (R³
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Area is the measure of how much exposed area any two dimensional object has. It is expressed in square units, and is calculated by adding together the areas of all the faces of the object.

Area formulas

Note: For 2D figures, the surface area and the area are the same.
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