Volume

Template:Alternateuses The amount of space occupied by an object is called the volume of the object. Every material object in the world occupies volume. Other meanings of the term are given below.

The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the 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:	$s^{3}=s\cdot s\cdot s$	s = length of a side
A rectangular prism:	$l\cdot w\cdot h$	l = length, w = width, h = height
A cylinder (circular prism):	$\pi r^{2}\cdot h$	r = radius of circular face, h = distance between faces
Any prism that has a constant cross sectional area along the height**:	$A\cdot h$	A = area of the base, h = height
A sphere:	${\frac {4}{3}}\pi r^{3}$	r = radius of sphere which is the first integral of the formula for Surface Area of a sphere
An ellipsoid:	${\frac {4}{3}}\pi abc$	a, b, c = semi-axes of ellipsoid
A pyramid:	${\frac {1}{3}}Ah$	A = area of base, h = height from base to apex
A cone (circular-based pyramid):	${\frac {1}{3}}\pi r^{2}h$	r = radius of circle at base, h = distance from base to tip
Any figure (calculus required)	$\int A(h)dh$	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.)

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 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

• Area
• Conversion of units
• Density
• Orders of magnitude (volume)
• Mass
• Pressure
• Ton (volume)

External links

• FORTRAN code for finding volumes of various shapes

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