# Volume: Wikis

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# Encyclopedia

The volume of any solid, liquid, gas, plasma, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as cubic meters, cubic centimeters, liters, or milliliters.

Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. More complicated shapes can be calculated by integral calculus if a formula exists for its boundary. The volume of any shape can be determined by displacement.

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.

Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure.

Conjugate variables
of thermodynamics
Pressure Volume
(Stress) (Strain)
Temperature Entropy
Chem. potential Particle no.

## Related terms

The density of an object is defined as mass per unit volume. The inverse of density is specific volume which is defined as volume divided by mass.

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 meters 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.

measure US Imperial metric
teaspoon 1/6 U.S. fluid ounce (about 4.929 mL) 1/6 Imperial fluid ounce (about 4.736 mL) 5 mL
tablespoon = 3 teaspoons ½ U.S. fluid ounce (about 14.79 mL) ½ Imperial fluid ounce (about 14.21 mL) 15 mL
cup 8 U.S. fluid ounces or ½ U.S. liquid pint (about 237 mL) 8 Imperial fluid ounces or 2/5 fluid pint (about 227 mL) 250 mL

In the UK, a tablespoon can also be five fluidrams (about 17.76 mL).

## Volume formulas

Shape Equation Variables
A cube $a^3\;$ a = length of any side (or edge)
A rectangular prism: $l \cdot w \cdot h$ l = length, w = width, h = height
A cylinder: $\pi r^2 h\;$ r = radius of circular face, h = height
A general prism: $B \cdot h$ B = area of the base, h = height
A sphere: $\frac{4}{3} \pi r^3$ r = radius of sphere
which is the integral of the Surface Area of a sphere
An ellipsoid: $\frac{4}{3} \pi abc$ a, b, c = semi-axes of ellipsoid
A pyramid: $\frac{1}{3}Bh$ B = area of the base, h = height of pyramid
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 if its cross-sectional area can be determined from h (no matter if the prism is slanted or the cross-sections change shape).

The units of volume depend on the units of length. If the lengths are in meters, the volume will be in cubic meters.

For their volume formulas, see the articles on tetrahedron and parallelepiped.

## Volume formula derivation

### Sphere

The volume of a sphere is the integral of infinitesimal circular slabs of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.

The radius of the circular slabs is $y = \sqrt{r^2-x^2}$

The surface area of the circular slab is πy2.

The volume of the sphere can be calculated as $\int_{-r}^r \pi(r^2-x^2) \,dx = \int_{-r}^r \pi r^2\,dx - \int_{-r}^r \pi x^2 \,dx$

Now $\int_{-r}^r \pi r^2\,dx = 2\pi r^3$
and $\int_{-r}^r \pi x^2 \,dx = 2 \pi \frac{r^3}{3}$

Combining yields $\left(2-\frac{2}{3}\right)\pi r^3 = \frac{4}{3}\pi r^3$

This formula can be derived more quickly using the formula for the sphere's surface area, which is r2. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to

$\int_0^r 4\pi r^2 \,dr$ = $\frac{4}{3}\pi r^3$

### Cone

The volume of a cone is the integral of infinitesimal circular slabs of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0) with radius r is as follows.

The radius of each circular slab is $\begin{cases} r, & \mbox{if }\mbox{ x=0} \\ 0, & \mbox{if }\mbox{ x=h} \end{cases}$, and varying linearly in between—that is, $r\frac{(h-x)}{h}$

The surface area of the circular slab is then $\pi \left(r\frac{(h-x)}{h}\right)^2 = \pi r^2\frac{(h-x)^2}{h^2}$

The volume of the cone can then be calculated as $\int_{0}^h \pi r^2\frac{(h-x)^2}{h^2} dx$

And after extraction of the constants: $\frac{\pi r^2}{h^2} \int_{0}^h (h-x)^2 dx$

Integrating gives us $\frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right) = \frac{1}{3}\pi r^2 h$

Volume is how much three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains,[1] often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.

Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.[2]

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

## Related terms

The density of an object is defined as mass per unit volume. The inverse of density is specific volume which is defined as volume divided by mass.

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 meters or its derived units).

Volume and capacity are also distinguished in capacity management, where capacity is defined as volume over a specified time period.

## Volume formulas

Shape Volume formula Variables
Any figure
(calculus required)
$V=\int A\left(h\right) \,dh$ 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 if its cross-sectional area can be determined from h).
Cube $a^3\;$ a = length of any side (or edge)
Cylinder $\pi r^2 h\;$ r = radius of circular face, h = height
Prism $B \cdot h$ B = area of the base, h = height
Rectangular prism $l \cdot w \cdot h$ l = length, w = width, h = height
Sphere $\frac\left\{4\right\}\left\{3\right\} \pi r^3$ r = radius of sphere
which is the integral of the Surface Area of a sphere
Ellipsoid $\frac\left\{4\right\}\left\{3\right\} \pi abc$ a, b, c = semi-axes of ellipsoid
Pyramid $\frac\left\{1\right\}\left\{3\right\}Bh$ B = area of the base, h = height of pyramid
Cone $\frac\left\{1\right\}\left\{3\right\} \pi r^2 h$ r = radius of circle at base, h = distance from base to tip
Tetrahedron[3] $\left\{\sqrt\left\{2\right\}\over12\right\}a^3 \,$ edge length $a$
Parallelpiped 

V= a b c \sqrt{K}
\begin\left\{align\right\}

K =& 1+2\cos(\alpha)\cos(\beta)\cos(\gamma) \\
& - \cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)


\end{align}

a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges

The units of volume depend on the units of length. If the lengths are in meters, the volume will be in cubic meters.

## Volume formula derivations

### Sphere

The volume of a sphere is the integral of infinitesimal circular slabs of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.

The radius of the circular slabs is $r = \sqrt\left\{y^2+x^2\right\}$

The surface area of the circular slab is $\pi r^2$.

The volume of the sphere can be calculated as $\int_\left\{-r\right\}^r \pi\left(y^2+x^2\right) \,dx = \int_\left\{-r\right\}^r \pi y^2\,dy + \int_\left\{-r\right\}^r \pi x^2 \,dx.$

Now $\int_\left\{-r\right\}^r \pi y^2\,dy = 2 \pi \frac\left\{r^3\right\}\left\{3\right\}$ and $\int_\left\{-r\right\}^r \pi x^2 \,dx = 2 \pi \frac\left\{r^3\right\}\left\{3\right\}.$

Combining yields $\left\left(\frac\left\{2\right\}\left\{3\right\}+\frac\left\{2\right\}\left\{3\right\}\right\right)\pi r^3 = \frac\left\{4\right\}\left\{3\right\}\pi r^3.$

This formula can be derived more quickly using the formula for the sphere's surface area, which is $4\pi r^2$. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to

$\int_0^r 4\pi u^2 \,du$ = $\frac\left\{4\right\}\left\{3\right\}\pi r^3.$

### Cone

The volume of a cone is the integral of infinitesimal circular slabs of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0) with radius r is as follows.

The radius of each circular slab is r if x = 0 and 0 if x = h, and varying linearly in between—that is, $r\frac\left\{\left(h-x\right)\right\}\left\{h\right\}.$

The surface area of the circular slab is then $\pi \left\left(r\frac\left\{\left(h-x\right)\right\}\left\{h\right\}\right\right)^2 = \pi r^2\frac\left\{\left(h-x\right)^2\right\}\left\{h^2\right\}.$

The volume of the cone can then be calculated as $\int_\left\{0\right\}^h \pi r^2\frac\left\{\left(h-x\right)^2\right\}\left\{h^2\right\} dx,$

and after extraction of the constants: $\frac\left\{\pi r^2\right\}\left\{h^2\right\} \int_\left\{0\right\}^h \left(h-x\right)^2 dx$

Integrating gives us $\frac\left\{\pi r^2\right\}\left\{h^2\right\}\left\left(\frac\left\{h^3\right\}\left\{3\right\}\right\right) = \frac\left\{1\right\}\left\{3\right\}\pi r^2 h.$

measure US Imperial metric
teaspoon 1/6 U.S. fluid ounce
1/6 Imperial fluid ounce
5 mL
tablespoon = 3 teaspoons ½ U.S. fluid ounce
½ Imperial fluid ounce
15 mL
cup 8 U.S. fluid ounces
or ½ U.S. liquid pint
10 Imperial fluid ounces
or ½ Imperial pint
250 mL

In the UK, a tablespoon can also be five fluidrams (about 17.76 mL). In Australia, a tablespoon is 4 teaspoons (20 mL).

## References

1. ^ "Your Dictionary entry for "volume"". Retrieved 2010-05-01.
2. ^ One litre of sugar (about 970 grams) can dissolve in 0.6 litres of hot water, producing a total volume of less than one litre. "Solubility". Retrieved 2010-05-01. "Up to 1800 grams of sucrose can dissolve in a liter of water."
3. ^ Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).

# Wiktionary

Up to date as of January 15, 2010
(Redirected to volume article)

### Definition from Wiktionary, a free dictionary

Wikipedia has articles on:

Wikipedia

## English

Wikipedia has an article on:

Wikipedia

### Etymology

From Old French volume, from Latin volūmen (book, roll), from volvō (roll, turn about).

### Pronunciation

• (UK) IPA: /ˈvɒl.juːm/, /ˈvɒl.jʊm/
• (US) IPA: /ˈvɑl.jum/, /ˈvɑl.jəm/
• help, file

### Noun

 Singular volume Plural volumes

volume (plural volumes)

1. A unit of three dimensional measure of space that comprises a length, a width and a height. It is measured in units of cubic centimeters in metric, cubic inches or cubic feet in English measurement. (The room is 9x12x8, so its volume is 864 cubic feet.)
2. Strength of sound. Measured in decibels. (Please turn down the volume on the stereo.)
3. The issues of a periodical over a period of one year. (I looked at this week's copy of the magazine. It was volume 23, issue 45.)
4. A single book of a publication issued in multi-book format, such as an encyclopedia. (The letter "G" was found in volume 4.)
5. A synonym for quantity. (The volume of ticket sales decreased this week.)
6. (economics) The total supply of money in circulation or, less frequently, total amount of credit extended, within a specified national market or worldwide.
7. (computing) An accessible storage area with a single file system, typically resident on a single partition of a hard disk.

cubic distance
sound

#### Translations

The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers. Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

## Dutch

Dutch Wikipedia has an article on:
Volume

Wikipedia nl

### Noun

volume n. (plural volumen or volumes)

## French

### Etymology

From Latin volūmen.

### Pronunciation

• IPA: /vɔ.lym/, SAMPA: /vO.lym/
• help, file

### Noun

volume m. (plural volumes)

• volumétrique
• volumineux

## Galician

### Etymology

From Latin volūmen (a book, roll).

### Noun

volume m. (plural volumes)

1. volume (quantity of space)
2. volume (single book of a published work)

## Italian

### Noun

volume m. (plural volumi)

# Simple English

The volume of an object describes how much physical space it takes up using the three dimensions of width, depth, and height.

## Use

The word volume can be used to describe real things like boxes, lakes, and buildings. All of these things have width, depth, and height. The formula for measuring volume is width×depth×height.