Difference between revisions of "Volume" - New World Encyclopedia

From New World Encyclopedia
Line 59: Line 59:
 
|}
 
|}
  
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)·|[[determinant|det]]('''a'''−'''b''', '''b'''−'''c''', '''c'''−'''d''')|, or any other combination of pairs of vertices that form a simply connected [[graph theory|graph]].
 
 
The volume of any [[tetrahedron]], given its vertices '''a''', '''b''', '''c''' and '''d''', is (1/6)·|[[determinant|det]]('''a'''−'''b''', '''b'''−'''c''', '''c'''−'''d''')|, or any other combination of pairs of vertices that form a simply connected [[graph theory|graph]].
 
  
 
==Volume measures: USA==
 
==Volume measures: USA==
  
 
[[U.S. customary units]]* of volume include the following:
 
[[U.S. customary units]]* of volume include the following:
 +
 
*U.S. fluid [[ounce]], about 29.6 mL
 
*U.S. fluid [[ounce]], about 29.6 mL
 
*U.S. liquid [[pint]] = 16 fluid ounces, or about 473 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. 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. liquid [[quart]] = 32 fluid ounces (two U.S. pints), or about 946 mL
32 fluid ounces or two U.S. pints, or about 950 mL
 
 
*U.S. dry quart = 1/32 U.S. bushel, or about 1.101 L
 
*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. 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 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 <!-- I wonder how customary this is. Isn't it officially being phased out? —>
+
*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 (water)|reservoir]] or an [[aquifer]]. It is the volume of water that would cover an [[area]] of one [[acre]] to a depth of one [[foot (unit of length)|foot]]. It is equivalent to 43,560 cubic feet or exactly 1233.481 837 547 52 m³.
 
  
*[[cubic inch]] = 16.387 064 cm<sup>3</sup>
+
*[[cubic inch]]* = 16.387 064 cm<sup>3</sup>
*[[cubic foot]] = 1,728 in<sup>3</sup> ≈ 28.317 dm<sup>3</sup>
+
*[[cubic foot]]* = 1,728 in<sup>3</sup> ≈ 28.317 dm<sup>3</sup>
*[[cubic yard]] = 27 ft<sup>3</sup> ≈ 0.7646 m<sup>3</sup>
+
*[[cubic yard]]* = 27 ft<sup>3</sup> ≈ 0.7646 m<sup>3</sup>
*[[cubic mile]] = 5,451,776,000 yd<sup>3</sup> = 3,379,200 acre-feet ≈ 4.168 km<sup>3</sup>
+
*[[cubic mile]]* = 5,451,776,000 yd<sup>3</sup> = 3,379,200 acre-feet ≈ 4.168 km<sup>3</sup>
 +
 
 +
The '''acre foot''' is often used in measuring the volume of water in a [[reservoir (water)|reservoir]] or [[aquifer]]. It is the volume of water that would cover an [[area]] of one [[acre]] to a depth of one [[foot (unit of length)|foot]]*. It is equivalent to 43,560 cubic feet or exactly 1233.481 837 547 52 m³.
  
 
==Volume measures: UK==
 
==Volume measures: UK==
  
The United Kingdom is undergoing [[metrication]] and is increasingly using the [[International System of Units |SI metric system's]] units of volume, that is, [[cubic meter]]* and [[liter]]. However, some former units of volume are still in varying degrees of usage:
+
The United Kingdom is increasingly using units of volume according to the [[International System of Units |SI metric system]], namely, the [[cubic meter]]* and [[liter]]. However, some former units of volume are still being used in varying degrees.
  
 
[[Imperial units]]* of volume:
 
[[Imperial units]]* of volume:
Line 91: Line 90:
 
*UK [[quart]] = 40 ounces or two pints1.137 L
 
*UK [[quart]] = 40 ounces or two pints1.137 L
 
*UK [[gallon]] = 4 quarts, or exactly 4.546 09 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 sold in SI units) and for milk (this too is increasingly being sold in SI units).
 
  
 
==Volume measures: cooking==
 
==Volume measures: cooking==
  
 
Traditional cooking measures for volume also include:
 
Traditional cooking measures for volume also include:
*[[teaspoon]] = 1/6 U.S. fluid ounce (about 4.929 mL)
+
 
 +
*[[teaspoon]]* = 1/6 U.S. fluid ounce (about 4.929 mL)
 
*teaspoon = 1/6 Imperial fluid ounce (about 4.736 mL) (Canada)
 
*teaspoon = 1/6 Imperial fluid ounce (about 4.736 mL) (Canada)
 
*teaspoon = 5 mL (metric)
 
*teaspoon = 5 mL (metric)
*[[tablespoon]] = ½ U.S. fluid ounce or 3 teaspoons (about 14.79 mL)
+
*[[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) (Canada)
 
*tablespoon = ½ Imperial fluid ounce or 3 teaspoons (about 14.21 mL) (Canada)
 
*tablespoon = 15 mL or 3 teaspoons (metric)
 
*tablespoon = 15 mL or 3 teaspoons (metric)
*tablespoon = 5 [[fluidram]]s (about 17.76 mL) (British)
+
*tablespoon = 5 [[fluidram]]*s (about 17.76 mL) (British)
*[[Cup (unit)|cup]] = 8 U.S. fluid ounces or ½ U.S. liquid pint (about 237 mL)
+
*[[Cup (unit)|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) (Canada)
 
*cup = 8 Imperial fluid ounces or ½ fluid pint (about 227 mL) (Canada)
 
*cup = 250 mL (metric)
 
*cup = 250 mL (metric)
Line 116: Line 114:
 
== Orders of magnitude (volume) ==
 
== Orders of magnitude (volume) ==
  
(Note: ''dam³'' and ''hm³'' stand for ''cubic decametre'' and ''cubic hectometre'' respectively. The terms in the left-hand column are common terminology.)
+
(Note: ''dam³'' and ''hm³'' stand for ''cubic decametre'' and ''cubic hectometre'' respectively.)
  
 
{| class="wikitable"
 
{| class="wikitable"

Revision as of 18:52, 19 December 2006

Template:Alternateuses The amount of space occupied by an object or system is called the volume of the object or system. The volume of an object is one of the physical properties of the object. (For other meanings of the term, see Additional meanings of "volume" below.)

The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. A one-dimensional object, such as a line in mathematics, or a two-dimensional object, such as a square, is assigned zero volume in three-dimensional space.

Mathematically, the volume of a body may be defined by means of integral calculus. In this approach, the volume of the body is taken to be approximately equal to the sum of volumes of a large number of small cubes or concentric cylindrical shells, and adding the individual volumes of those shapes. In differential geometry, volume is expressed by means of the "volume form."

Volume is sometimes distinguished from the capacity of a container. The term capacity is used to indicate how much a container can hold (commonly measured in liters or its derived units), and volume indicates how much space the object displaces (commonly measured in cubic meters or its derived units). Alternatively, in a capacity management setting, capacity is defined as volume over a specified time period.

Volume is a fundamental parameter in thermodynamics and is regarded as a "conjugate variable" to pressure.

Conjugate variables
of thermodynamics
Pressure Volume
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 = distance between faces
Any prism that has a constant cross sectional area along the height**: A = area of the base, h = height
A sphere: r = radius of sphere
which is the first integral of the formula for Surface Area of a sphere
An ellipsoid: a, b, c = semi-axes of ellipsoid
A pyramid: A = area of base, h = height from base to apex
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 (no matter if the prism is slanted or the cross-sections change shape).

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

Volume measures: USA

U.S. customary units of volume include the following:

  • 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 (two U.S. pints), 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
  • cubic inch = 16.387 064 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

The acre foot is often used in measuring the volume of water in a reservoir or 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.

Volume measures: UK

The United Kingdom is increasingly using units of volume according to the SI metric system, namely, the cubic meter and liter. However, some former units of volume are still being used in varying degrees.

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

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) (Canada)
  • 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) (Canada)
  • 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) (Canada)
  • cup = 250 mL (metric)

Relationship to density

The volume of an object is equal to its mass divided by its average density. (The term "average density" is used for an object that does not have uniform 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, expressed in units such as cubic meters per kilogram (m³·kg-1). It is the reciprocal of density.

Orders of magnitude (volume)

(Note: dam³ and hm³ stand for cubic decametre and cubic hectometre respectively.)

List of orders of magnitude for volume
Factor (m³) Multiple Value
10−105 4×10−105 m³ is the Planck volume
10−45 Volume of a proton
10−33 Volume of a hydrogen atom (6.54×10-32 m³) 10-33, 10-32, 10-31
10−21 1 attolitre Volume of a typical virus (5 attolitres) 10-21, 10-20, 10-19
10−15 1 picolitre A small grain of sand (0.063 mm diameter, 3 micrograms, 130 picolitres) 10-15, 10-14, 10-13
10−12 1 nanolitre A medium grain of sand (0.5 mm diameter, 1.5 milligrams, 62 nanolitres) 10-12, 10-11, 10-10
10−9 1 microlitre A large grain of sand (2.0 mm diameter, 95 milligrams, 4 microlitres) 10-9, 10-8, 10-7
10−6 1 millilitre
(1 cubic centimetre)
1 teaspoon = 3.55 ml to 5 ml

1 tablespoon = 14.2 ml to 20 ml

1 cm³, 10 cm³, 100 cm³


10−3 1 litre
(1 cubic decimetre)
1 U.S. quart = 0.95 liters;
1 United Kingdom quart = 1.14 litres
1 dm³, 10 dm³, 100 dm³
100 1000 litres Fuel tank for a 12-passenger turboprop airplane. 1 m³, 10 m³, 100 m³
103 1000 cubic metres
(1 million litres)
A medium-size forest pond.
An Olympic size swimming pool, 25 metres by 50 metres by 2 metres deep, holds at least 2.5 megalitres.
1 dam³, 10 dam³, 100 dam³
106 1 million cubic metres 1 hm³, 10 hm³, 100 hm³
109 1 cubic kilometre Volume of Lake Mead (Hoover Dam) = 35.2 km3

Volume of crude oil on Earth = ~300 km3

1 km³, 10 km³, 100 km³
1012 1000 cubic kilometres Volume of Lake Superior = 12,232 km3 1012, 1013, 1014
1015 1015, 1016, 1017
1018 Volume of water in all Earth oceans = 1.3×1018  1018, 1019, 1020
1021 Volume of Earth = ~1×1021  1021, 1022, 1023
1024 Volume of Jupiter = ~1×1025  1024, 1025, 1026
1027 Volume of Sun = ~1×1027  1027, 1028, 1029
1030 Volume of a red giant the same mass as the Sun = ~5×1032  1030, 1031, 1032
1033 Volume of Betelgeuse = ~2.75×1035  1033, 1034, 1035
1054 Volume of small dwarf galaxy like NGC 1705 = ~3×1055  1054, 1055, 1056
1057 Volume of dwarf galaxy like the Large Magellanic Cloud = ~3×1058  1057, 1058, 1059
1060 Volume of galaxy like the Milky Way = ~3.3×1061  1060, 1061, 1062
1066 Volume of the Local Group = ~5×1068  1066, 1067, 1068
1072 Volume of the Virgo Supercluster = ~4×1073  1072, 1073, 1074
1081 Approximate volume of the observable universe 1.6×1081 m³

Additional meanings of "volume"

Besides the above meaning, the term "volume" can refer to the following concepts:

  • Volume form in mathematics.
  • Loudness, in acoustics. It is related to:
    • Amplitude of the sound wave
    • Sound pressure level
    • Dynamics, in music.
  • Quantity, as in "the volume of ticket sales."
  • Volume, in computing, is a storage area with a single file system, typically residing on a single partition of a hard disk.
  • Volume is a term in data compression, for a file that has been compressed and split into different parts.

See also

External links

Credits

New World Encyclopedia writers and editors rewrote and completed the Wikipedia article in accordance with New World Encyclopedia standards. This article abides by terms of the Creative Commons CC-by-sa 3.0 License (CC-by-sa), which may be used and disseminated with proper attribution. Credit is due under the terms of this license that can reference both the New World Encyclopedia contributors and the selfless volunteer contributors of the Wikimedia Foundation. To cite this article click here for a list of acceptable citing formats.The history of earlier contributions by wikipedians is accessible to researchers here:

The history of this article since it was imported to New World Encyclopedia:

Note: Some restrictions may apply to use of individual images which are separately licensed.