# Volume

Volume, also called capacity, is a quantification of how much space an object occupies. The international unit for volume is the cubic meter. Template:Conjugate variables (thermodynamics) 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, and adding the volumes of those cubes. 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 is a fundamental parameter in thermodynamics and it is conjugate to pressure.

## Volume formula

Common equations for volume:

A cube:
$\displaystyle s^3 = s \cdot s \cdot s$ (where s is the length of a side)

A rectangular prism:
$\displaystyle l \cdot w \cdot h$ (length, width, height)

A cylinder:
$\displaystyle \pi \cdot r^2 h$ (r = radius of circular face, h = distance between faces)

A sphere:
$\displaystyle \frac{4}{3} \pi r^3$ (r = radius of sphere) - (which is the first intergal of the formula for Surface Area of a sphere

An ellipsoid:
$\displaystyle \frac{4}{3} \pi abc$ (a, b, c = semi-axes of ellipsoid)

A pyramid:
$\displaystyle \frac{1}{3} A h$ (A = area of base, h = height from base to apex)

A cone (circular-based pyramid):
$\displaystyle \frac{1}{3} \pi r^2 h$ (r = radius of circle at base, h = distance from base to tip)

Any prism that has a constant cross sectional area along the height**:
$\displaystyle A \cdot h$ (A = area of the base, h = height)

Any figure (calculus required)
$\displaystyle \int A(h) dh$

where h is any dimension of the figure, and A(h) is the 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(ab, bc, cd)|, or any other combination of pairs of vertices that form a simply connected graph.

## Volume measures: other metric units

A commonly used metric unit for volume is the litre (American spelling liter), and one thousand litres is the volume of a cubic metre (American spelling cubic meter), which was formerly termed a stere and often called a "cube" in engineering slang. A cubic centimetre (American spelling cubic centimeter) is the same volume as a millilitre.

## 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 two U.S. pints, or about 946 mL
• U.S. dry quart = 1/32 U.S. bushel, or about 1.101 L
• U.S. 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

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 pints, or about 1.137 L
• UK gallon = 160 ounces or four quarts, or exactly 4.546 09 L

May it be noted that due to metrication within the UK, the quart is now obsolete and the fluid ounce extremely rare. The gallon is only used for transportation uses, (it is illegal for petrol & 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 & cider (bottled & canned beer is sold in SI units) and for milk (this too is increasingly being sold in SI units).

## Volume measures: cooking

Traditional cooking measures for volume also include:

• teaspoon = 1/6 U.S. fluid ounce (about 4.929 mL)
• teaspoon = 5 mL (metric)
• tablespoon = 1/2 U.S. fluid ounce or 3 teaspoons (about 14.79 mL)
• tablespoon = 1/2 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 1/2 U.S. liquid pint (about 237 mL)
• cup = 8 Imperial fluid ounces or 1/2 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).

## Volume comparisons

To help compare different volumes, see orders of magnitude (volume)