Volume

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If each side of a cube has length s, the volume of the cube is equal to s³.

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 given a numerical value that quantifies the amount of three-dimensional 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. In thermodynamics, volume is a fundamental parameter that is regarded as a "conjugate variable" to pressure.

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.

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 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.

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 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 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

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.

External links

FORTRAN code for finding volumes of various shapes

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