Volume 32498 222286587 2008-06-28T14:50:44Z Hydraton31 1220118 Removed random text (possibly a test edit) and fixed wiki-formatting. {{alternateuses}} The '''volume''' of any solid, liquid, or gas is how much three-[[dimension]]al space it occupies, often quantified numerically. One-dimensional figures (such as [[line (mathematics)|lines]]) and two-dimensional shapes (such as [[square (geometry)|square]]s) are assigned zero volume in the three-dimensional space. Volumes of straight-edged and circular shapes are calculated using arithmetic formulae.<!--- "formulae" is a correct plural of "formula" ---> Volumes of other curved shapes are calculated using [[integral calculus]], by approximating the given body with a large amount of small [[cube (geometry)|cubes]] or concentric [[cylinder (geometry)|cylindrical]] shells, and adding the individual volumes of those shapes. The volume of irregularly shaped objects can be determined by [[Displacement (fluid)|displacement]]. If an irregularly shaped object is less dense than the fluid, you will need a weight to attach to the floating object. A sufficient weight will cause the object to sink. The final volume of the unknown object can be found by subtracting the volume of the attached heavy object and the total fluid volume displaced. The generalization of volume to arbitrarily many dimensions is called [[content (measure theory)| content]].{{Fact|date=February 2007}} 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 [[litre]]s or its derived units), and volume being how much space an object displaces (commonly measured in [[cubic metre]]s or its derived units).<!--- a 2 litre bowl does not (necessarily anyway) displace 2 litres of air (or water)--> The volume of a dispersed gas is the capacity of its container. If more gas is added to a closed container, the container either expands (as in a balloon) or the [[pressure]] inside the container increases. 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 variables (thermodynamics)|conjugate]] to [[pressure]]. {{Conjugate variables (thermodynamics)}} == Volume formulas == <!--- "formulas" is a correct plural of "formula" ---> {| class=prettytable |- ! colspan = 3 | Common [[equation]]s for volume: |- !Shape !Equation !Variables |- |A [[Cube (geometry)|cube]]: |<math>s^3</math> |''s'' = length of any side |- |A rectangular [[Prism (geometry)|prism]]: |<math>l \cdot w \cdot h</math> |l = ''l''ength, w = ''w''idth, h = ''h''eight |- |A [[cylinder (geometry)|cylinder]] (circular prism): |<math>\pi r^2 h</math> |''r'' = radius of circular face, ''h'' = height |- |Any prism that has a constant cross sectional area along the height**: |<math>A \cdot h</math> |''A'' = area of the base, ''h'' = height |- |A [[sphere]]: |<math>\frac{4}{3} \pi r^3</math> |''r'' = radius of sphere<br>which is the [[integral]] of the [[Surface Area]] of a [[sphere]] |- |An [[ellipsoid]]: |<math>\frac{4}{3} \pi abc</math> |''a'', ''b'', ''c'' = semi-axes of ellipsoid |- |A [[Pyramid (geometry)|pyramid]]: |<math>\frac{1}{3}Ah</math> |''A'' = area of the base, ''h'' = height of pyramid |- |A [[Cone (geometry)|cone]] (circular-based pyramid): |<math>\frac{1}{3} \pi r^2 h</math> |''r'' = radius of [[circle]] at base, ''h'' = distance from base to tip |- |Any figure ([[integral calculus|calculus]] required) |<math>\int A(h) \,dh</math> |''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)</small>. ^* |} (The units of volume depend on the units of length - if the lengths are in meters, the volume will be in cubic '''meters''', etc) 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)·|[[determinant|det]]('''a'''&minus;'''b''', '''b'''&minus;'''c''', '''c'''&minus;'''d''')|, or any other combination of pairs of vertices that form a simply connected [[graph theory|graph]]. == 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) *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) *tablespoon = 15 mL or 3 teaspoons (metric) *tablespoon = 5 [[fluidram]]s (about 17.76 mL) (British) *[[Cup (unit)|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) *cup = 250 mL (metric) ==Relationship to density== The [[density]] of an object is defined as mass per unit volume. The term ''[[specific volume]]'' is used for volume divided by mass. This is the [[Reciprocal (mathematics)|reciprocal]] of the [[mass density]], expressed in units such as cubic meters per kilogram (m³·kg<sup>-1</sup>). == Volume formula derivation == {| class=prettytable |- !Shape !Volume formula derivation |- |[[Sphere]] |The volume of a sphere is the [[integral]] of infinitesimal circular slabs of width <math>dx</math>. The calculation for the volume of a sphere with center 0 and radius r is as follows. <br/>The radius of the circular slabs is <math> y = \sqrt{r^2-x^2}</math> <br/>The surface of the circular slab is <math> \pi \cdot y^2 </math> <br/>The volume of the sphere can be calculated as <math> \int_{-r}^r \pi(r^2-x^2) \,dx</math> <br/>Replacing <math>x</math> by <math>x \cdot r</math>, so that the integral boundaries become -1 and +1, we get <math> \pi r^3 \int_{-1}^1 (1-x^2) \,dx</math> <br/>The [[antiderivative]] needed can be determined very easily as <math> x-\frac{x^3}{3}</math> <br/>Thus, the sphere volume amounts to V<sub>sphere</sub> = <math> \pi r^3 \cdot[1-1/3-(-1+1/3)] </math> = <math> \frac{4}{3}\pi r^3</math> <br/><br/> This formula can be derived more quickly using the formula for the sphere [[surface area]], which is <math>4\pi r^2</math>. The volume of the sphere consists of layers of infinitesimal spherical slabs, and THE sphere volume is equal to <math> \int_0^r 4\pi r^2 \,dr</math> = <math> \frac{4}{3}\pi r^3</math> |} == See also == <div class="references-small" style="-moz-column-count:3; column-count:3;"> *[[Area]] *[[Conversion of units#Volume|Conversion of units]] *[[Density]] *[[Orders of magnitude (volume)]] *[[Length]] *[[Mass]] *[[Ton (volume)]] *[[Weight]] *[[Dimensioning]] *[[Dimensional weight]] </div> ==External links== * [http://www.phy.ilstu.edu/~mnorton/Geometry.txt FORTRAN code for finding volumes of various shapes] * [http://www.pneumofore.com/support/tools Unit Converter Tool also for Volume] [[Category:Volume|*]] [[af:Volume]] [[bg:Обем]] [[ca:Volum]] [[cs:Objem]] [[da:Rumfang]] [[de:Volumen]] [[el:Όγκος]] [[es:Unidades de volumen]] [[eo:Volumeno]] [[eu:Bolumen (espazioa)]] [[fr:Volume]] [[hi:आयतन]] [[ko:부피]] [[hr:Obujam]] [[id:Volume]] [[it:Volume]] [[he:נפח]] [[lb:Volumen]] [[lt:Tūris]] [[hu:Térfogat]] [[mk:Волумен]] [[ms:Isi padu]] [[nl:Inhoud]] [[ja:体積]] [[no:Volum]] [[nn:Volum]] [[pl:Objętość (matematyka)]] [[pt:Volume]] [[ro:Volum]] [[qu:P'ulin]] [[ru:Объём]] [[simple:Volume]] [[sk:Objem]] [[sl:Prostornina]] [[fi:Tilavuus]] [[sv:Volym (geometri)]] [[ta:கனவளவு]] [[vi:Thể tích]] [[tr:Hacim]] [[uk:Об'єм]] [[zh:体积]]