The volume of any solid, liquid, gas, plasma, or vacuum is how much threedimensional space it occupies, often quantified numerically. Onedimensional figures (such as lines) and twodimensional shapes (such as squares) are assigned zero volume in the threedimensional space. Volume is commonly presented in units such as cubic meters, cubic centimeters, liters, or milliliters.
Volumes of some simple shapes, such as regular, straightedged, 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. 
Contents 
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).
Shape  Equation  Variables 

A cube  a = length of any side (or edge)  
A rectangular prism:  l = length, w = width, h = height  
A cylinder:  r = radius of circular face, h = height  
A general prism:  B = area of the base, h = height  
A sphere:  r = radius of sphere which is the integral of the Surface Area of a sphere 

An ellipsoid:  a, b, c = semiaxes of ellipsoid  
A pyramid:  B = area of the base, h = height of pyramid  
A cone (circularbased pyramid):  r = radius of circle at base, h = distance from base to tip  
Any figure (calculus required)  h = any dimension of the figure, A(h) = area of the crosssections perpendicular to h described as a function of the position along h. This will work for any figure if its crosssectional area can be determined from h (no matter if the prism is slanted or the crosssections 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.
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
The surface area of the circular slab is πy^{2}.
The volume of the sphere can be calculated as
Combining yields
This formula can be derived more quickly using the formula for the sphere's surface area, which is 4πr^{2}. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to
=
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 , and varying linearly in between—that is,
The surface area of the circular slab is then
The volume of the cone can then be calculated as
And after extraction of the constants:
Integrating gives us
Volume is how much threedimensional 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, straightedged, 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. Onedimensional figures (such as lines) and twodimensional shapes (such as squares) are assigned zero volume in the threedimensional 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.
Contents 
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.
Shape  Volume formula  Variables 

Any figure (calculus required)  $V=\backslash int\; A(h)\; \backslash ,dh$  h = any dimension of the figure, A(h) = area of the crosssections perpendicular to h described as a function of the position along h. (This will work for any figure if its crosssectional area can be determined from h). 
Cube  $a^3\backslash ;$  a = length of any side (or edge) 
Cylinder  $\backslash pi\; r^2\; h\backslash ;$  r = radius of circular face, h = height 
Prism  $B\; \backslash cdot\; h$  B = area of the base, h = height 
Rectangular prism  $l\; \backslash cdot\; w\; \backslash cdot\; h$  l = length, w = width, h = height 
Sphere  $\backslash frac\{4\}\{3\}\; \backslash pi\; r^3$  r = radius of sphere which is the integral of the Surface Area of a sphere 
Ellipsoid  $\backslash frac\{4\}\{3\}\; \backslash pi\; abc$  a, b, c = semiaxes of ellipsoid 
Pyramid  $\backslash frac\{1\}\{3\}Bh$  B = area of the base, h = height of pyramid 
Cone  $\backslash frac\{1\}\{3\}\; \backslash pi\; r^2\; h$  r = radius of circle at base, h = distance from base to tip 
Tetrahedron^{[3]}  $\{\backslash sqrt\{2\}\backslash over12\}a^3\; \backslash ,$  edge length $a$ 
Parallelpiped  $$ V= a b c \sqrt{K}
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.
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 $r\; =\; \backslash sqrt\{y^2+x^2\}$
The surface area of the circular slab is $\backslash pi\; r^2$.
The volume of the sphere can be calculated as $\backslash int\_\{r\}^r\; \backslash pi(y^2+x^2)\; \backslash ,dx\; =\; \backslash int\_\{r\}^r\; \backslash pi\; y^2\backslash ,dy\; +\; \backslash int\_\{r\}^r\; \backslash pi\; x^2\; \backslash ,dx.$
Now $\backslash int\_\{r\}^r\; \backslash pi\; y^2\backslash ,dy\; =\; 2\; \backslash pi\; \backslash frac\{r^3\}\{3\}$ and $\backslash int\_\{r\}^r\; \backslash pi\; x^2\; \backslash ,dx\; =\; 2\; \backslash pi\; \backslash frac\{r^3\}\{3\}.$
Combining yields $\backslash left(\backslash frac\{2\}\{3\}+\backslash frac\{2\}\{3\}\backslash right)\backslash pi\; r^3\; =\; \backslash frac\{4\}\{3\}\backslash pi\; r^3.$
This formula can be derived more quickly using the formula for the sphere's surface area, which is $4\backslash pi\; r^2$. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to
$\backslash int\_0^r\; 4\backslash pi\; u^2\; \backslash ,du$ = $\backslash frac\{4\}\{3\}\backslash pi\; r^3.$
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, $r\backslash frac\{(hx)\}\{h\}.$
The surface area of the circular slab is then $\backslash pi\; \backslash left(r\backslash frac\{(hx)\}\{h\}\backslash right)^2\; =\; \backslash pi\; r^2\backslash frac\{(hx)^2\}\{h^2\}.$
The volume of the cone can then be calculated as $\backslash int\_\{0\}^h\; \backslash pi\; r^2\backslash frac\{(hx)^2\}\{h^2\}\; dx,$
and after extraction of the constants: $\backslash frac\{\backslash pi\; r^2\}\{h^2\}\; \backslash int\_\{0\}^h\; (hx)^2\; dx$
Integrating gives us $\backslash frac\{\backslash pi\; r^2\}\{h^2\}\backslash left(\backslash frac\{h^3\}\{3\}\backslash right)\; =\; \backslash frac\{1\}\{3\}\backslash pi\; r^2\; h.$
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)  10 Imperial fluid ounces or ½ Imperial pint (about 284 mL)  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).
The Wikibook Calculus has a page on the topic of 
The Wikibook Geometry has a page on the topic of 
Contents 
From Old French volume, from Latin volūmen (“‘book, roll’”), from volvō (“‘roll, turn about’”).
Singular 
Plural 
volume (plural volumes)




Wikipedia ^{nl}
volume n. (plural volumen or volumes)
volume m. (plural volumes)
From Latin volūmen (“‘a book, roll’”).
volume m. (plural volumes)
volume m. (plural volumi)
The volume of an object describes how much physical space it takes up using the three dimensions of width, depth, and height.
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.
