Volume
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, volumes are defined by means of 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 generalization of volume to arbitrarily many dimensions is called content.Template:Citeneeded 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 |
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 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
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. 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 = 16.387 064 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
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 liter. 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 sold in SI units) and for milk (this too is increasingly being sold in SI units). 1 cup
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. 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).
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.
Orders of magnitude (volume)
The pages linked in the right-hand column contain lists of volumes that are of the same order of magnitude (power of ten). Rows in the table represent increasing powers of a thousand. (Note: dam³ and hm³ stand for cubic decametre and cubic hectometre respectively. The terms in the left-hand column are common terminology.)
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 m³, 10-32 m³, 10-31 m³ |
10−21 | 1 attolitre | Volume of a typical virus (5 attolitres) | 10-21 m³, 10-20 m³, 10-19 m³ |
10−15 | 1 picolitre | A small grain of sand (0.063 mm diameter, 3 micrograms, 130 picolitres) | 10-15 m³, 10-14 m³, 10-13 m³ |
10−12 | 1 nanolitre | A medium grain of sand (0.5 mm diameter, 1.5 milligrams, 62 nanolitres) | 10-12 m³, 10-11 m³, 10-10 m³ |
10−9 | 1 microlitre | A large grain of sand (2.0 mm diameter, 95 milligrams, 4 microlitres) | 10-9 m³, 10-8 m³, 10-7 m³ |
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 m³, 1013 m³, 1014 m³ |
1015 | — | — | 1015 m³, 1016 m³, 1017 m³ |
1018 | — | Volume of water in all Earth oceans = 1.3×1018 | 1018 m³, 1019 m³, 1020 m³ |
1021 | — | Volume of Earth = ~1×1021 | 1021 m³, 1022 m³, 1023 m³ |
1024 | — | Volume of Jupiter = ~1×1025 | 1024 m³, 1025 m³, 1026 m³ |
1027 | — | Volume of Sun = ~1×1027 | 1027 m³, 1028 m³, 1029 m³ |
1030 | — | Volume of a red giant the same mass as the Sun = ~5×1032 | 1030 m³, 1031 m³, 1032 m³ |
1033 | — | Volume of Betelgeuse = ~2.75×1035 | 1033 m³, 1034 m³, 1035 m³ |
1054 | — | Volume of small dwarf galaxy like NGC 1705 = ~3×1055 | 1054 m³, 1055 m³, 1056 m³ |
1057 | — | Volume of dwarf galaxy like the Large Magellanic Cloud = ~3×1058 | 1057 m³, 1058 m³, 1059 m³ |
1060 | — | Volume of galaxy like the Milky Way = ~3.3×1061 | 1060 m³, 1061 m³, 1062 m³ |
1066 | — | Volume of the Local Group = ~5×1068 | 1066 m³, 1067 m³, 1068 m³ |
1072 | — | Volume of the Virgo Supercluster = ~4×1073 | 1072 m³, 1073 m³, 1074 m³ |
1081 | — | Approximate volume of the observable universe 1.6×1081 m³ |
See also
External links
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