# Volume

In Physical, the volume of an object measures “the extension in the space” which it has in the three directions at the same time, just as the surface of a figure in the plan measurement “the extension” as it has in the two directions at the same time.

Volume measures in Cubic meter in the international system. One frequently uses the liter, in particular for liquids.

Thus, one regards the volume as a extensive Grandeur and the intensive Grandeur associated Thermodynamique is the Pression.

In Mathematical, the volume of part of the space is its measurement. For the solid S simple (Parallelepiped and objects of revolution), there exist mathematical formulas making it possible to determine their volume according to their characteristic dimensions. In Euclidean Geometry, the volume of the Parallélépipède generated by 3 Vecteur S noncoplanar $\left(\ vec v_1, \ vec v_2, \ vec v_3\right)$ is calculated thanks to the mixed Produit of the three vectors: $V = |\ det \left(\ vec v_1, \ vec v_2, \ vec v_3\right)|$. Calculations of volume evolved/moved during the history while following progress of the Infinitesimal calculus. Thus the first volumes were calculated thanks to the Méthode of exhaustion, then by using the Principe of Cavalieri and to finish by calculating triple integrals.

## Units of volume

The unit of volume of the international system is the cubic meter (m ³) and its derivatives (DM ³, cm ³, mm ³). But of other units of volume persist especially in the Anglo-Saxon countries (see Conversion of the units)).

Volumes of liquid matter often have their own units (liter, pint, barrel). The installation of the metric system largely simplified the number of units of volume used which in the old mode counted some more than twenty (see Measuring units of the Old Mode).

For gases where one wants to know the quantity of matter (number of molecule) contained in a volume given whatever the pressure and the temperature, two definitions of correction exist:

• the cubic meter known as normal expressed in m3 (N) correspondent with a volume of gas brought back under a pressure of 1013,25 hPa (pressure of a standard atmosphere or 1 Atm) and a temperature of 0°C.
• the standard cubic meter said expressed in m3 (S) correspondent to a volume of gas brought back under a pressure of 1013,25 hPa (pressure of a standard atmosphere or 1 Atm) and a temperature of 15°C.
Volumes describes above correspond to volumes known as corrected. The volume which does not take account of these corrections is known as rough. One meets these volumes in the development of the flows (see Débit) and of the Calorific value of gases.

In the European Union, many volumes (and masses), on the consumables, are indicated in estimated Quantité. They are marked like such, of a “E” lower-case.

In mathematics, the unit of volume does not appear in the formulas. It is implicitly given by the volume of the cubic unit . If, for example, for questions of scale, the cubic unit has as an edge 2 cm, a volume of X ( cubic unit ) corresponds to 8X cm ³.

## Some formulas

In the continuation one will note
• $V$ volume
• $B$ and $b$ Grande Base and small base
• $H$: height (or outdistances separating the two faces)
• $D$ or $d$ the diameter
• $R$ or $r$ the ray
• $a$ the edge
• $L$ or $l$: the length and the width of a rectangle

### Solids of Plato

They are the five only regular polyhedrons. If the edge of the polyhedron is $a \,$, one has

• For the Tétraèdre: $V = \ frac \left\{1\right\} \left\{12\right\} \ sqrt \left\{2\right\} a^3$
• For the Cubic: $V = a^3 \,$
• For the Octahedral : $V = \ frac 13 \ sqrt 2 a^3$
• For the Dodecahedron: $V = \ frac 14 \left(15 + 7 \ sqrt 5\right) a^3$
• For the Icosaèdre: $V = \ frac 56 \ varphi^2a^3$ where $\ varphi$ is the Golden section

### The prism S and Cylinder S

The general formula is always: Surface of the base × Hauteur
• the right Prism: $V =B \ times H$

• the right-angled or paved Parallelepiped: $V = L \ times L \ times H$
• the Cylinder of revolution: $V = \ pi R^2H$

### The Pyramid S and cone S

The general formula is always: $V = \ frac 13 B \ times H$
• the circular cone: $V = \ frac \left\{\ pi\right\} \left\{3\right\} R^2H$

• the cone (or the pyramid) truncated (E) by a plan parallel with the base: $V= \left\{H \ over3\right\} \left(B+b+ \ sqrt \left\{Bb\right\}\right)$

### The swell

• the Sphere: $V = \left\{4 \ over 3\right\} \ pi R^3$ or $V = \ pi \left\{D^3 \ over 6\right\}$
• the segment of a sphere: $V = \ frac \left\{\ pi\right\} \left\{3\right\} H^2 \left(3R- H\right)$ where R is the ray of the ball and H the height of the cap.
• the bored sphere of a cylinder (napkin ring): $V = \ frac \left\{\ pi\right\} \left\{6\right\} H^3$
• the spherical sector (intersection between a cone of top O and the ball of center O: $V = \ frac 23 \ pi R^2H$ where H is the height of the cap and R the ray of the ball.

### Solids of revolution

The Théorème of Guldin (or regulates of Pappus) makes it possible to calculate the volume of a solid of revolution generated by the revolution of an element of surface S plane around an axis located in its plan and not cutting it, for little that one knows the Center of gravity G of the element of surface S.

$V = 2 \ pi R \ cdot S$ where R is the distance separating the point G from the axis of rotation.
This formula makes it possible to determine following volumes:
• the Torus: $V = 2 \ pi^2 Rr^2$ where R is the ray of the circle of center G turning around the axis $\left(\ Delta\right)$ and where R is the distance from G to $\left(\ Delta\right)$.
• the barrel: Kepler gives a formula approached for the volume of a barrel, which appears exact when the barrel is generated by a sphere, a pyramid, a Hyperboloïde with a tablecloth, a elliptic Paraboloïde, a Ellipsoïde of revolution. If $B_1$ and $B_2$ are surfaces of the bases and $B_3$ the sectional surface to middle height then
$V = \ frac h6 \left(B_1 + B_2 + 4B_3\right)$

### Others

• the Conoïde circular right (example the incisor): $V = \ frac 12 \ pi R^2H$ where R is the ray of the basic circle and H the height of the conoïde.
• the ingot (Hexahedron formed of two parallel rectangular bases and 4 trapezoidal side faces). One finds the formula of Kepler: $V = \ frac h6 \left(B_1+B_2+4B_3\right)$ where $B_1$ and $B_2$ are surfaces of the two rectangular bases and $B_3$ sectional surface to middle height.

## Volume and integral calculus

If $\ mathcal D$ is a limited part of $\ R^2$, the volume of the Cylindre having for generator the border of $\ mathcal D$, delimited by the $z=0$ plan and surfaces it equation $z=f \left(X, there\right)$ - with F positive and continuous on $\ mathcal D$ - is:

$V = \ iint_ \ mathcal D F \left(X, there\right) \, \ mathrm \left\{D\right\} X \, \ mathrm \left\{D\right\} y$
If the field $\ mathcal D$ is defined by simple conditions
$V = \ int_ \left\{x_1\right\} ^ \left\{x_2\right\} \! \ int_ \left\{y_1 \left(X\right)\right\}^ \left\{y_2 \left(X\right)\right\} F \left(X, there\right) \, \ mathrm \left\{D\right\} there \, \ mathrm \left\{D\right\} x$
If $\ mathcal A$ is a limited part of $\ R^3$ and if the constant function 1 is integrable on $\ mathcal A$, the volume of $\ mathcal A$ is then
$V = \ iiint _ \ mathcal has \ mathrm \left\{D\right\} X \, \ mathrm \left\{D\right\} there \, \ mathrm \left\{D\right\} z$
If the field $\ mathcal A$ is defined by simple conditions
$V = \ int_ \left\{z_1\right\} ^ \left\{z_2\right\} \! \ int_ \left\{y_1 \left(Z\right)\right\}^ \left\{y_2 \left(Z\right)\right\}\! \ int_ \left\{x_1 \left(Z, there\right)\right\}^ \left\{x_2 \left(Z, there\right)\right\}\ mathrm \left\{D\right\} X \, \ mathrm \left\{D\right\} there \, \ mathrm \left\{D\right\} z$
By linearity of integration, a field difficult to define can be partitionné in several under-fields exprimables them in simple conditions.

If the field $\ mathcal A$ is expressed better in cylindrical Coordonnées by simple conditions $\ mathcal A\text{'}$, calculation can be expressed by

$V = \ iiint _ \left\{\ mathcal A\text{'}\right\} R \, \ mathrm \left\{D\right\} R \, \ mathrm \left\{D\right\} \ theta \, dz$ where $\ mathcal A\text{'}$ is a limited part of $\ R_+ \ times \ times \ R$
If the field $\ mathcal A$s' expresses better in spherical Coordonnées by simple conditions $\ mathcal has$ , calculation can be expressed by
$V = \ iiint _ \left\{\ mathcal has$} r^2 \ sin (\ theta) \, \ mathrm {D} R \, \ mathrm {D} \ theta \, \ mathrm {D} \ phi where $\ mathcal has$ is a limited part of $\ R_+ \ times \ times$.
If the field $\ mathcal A$ is a solid of revolution whose border is generated by the rotation of a curve of equation there = F (X) around the axis (OX) , the calculation of volume is reduced to a simple integral
$V = \ pi \ int_ \left\{x_1\right\} ^ \left\{x_2\right\} f^2 \left(X\right) \, \ mathrm \left\{D\right\} x$
Lastly, the Théorème of Green-Ostrogradsky makes it possible to reduce the calculation of volume to a Intégrale of surface
$V = \ iiint _A \ mathrm \left\{D\right\} V = \ frac 13 \ iint_ \left\{\ share \ mathcal has\right\} \left(X, there, Z\right) \ vec N \, \ mathrm \left\{D\right\} S$
where $\ share \ mathcal A$ is the border of $\ mathcal A$, and $\ vec n$ the Unit vector normal with dS directed towards the outside of $\ mathcal A$.

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