Volume

For other uses, see Volume (disambiguation).

The volume of any solid, liquid, or gas is how much three-dimensional space it occupies, often quantified numerically. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

Volumes of straight-edged and circular shapes are calculated using arithmetic formulae. Volumes of other curved shapes are calculated using integral calculus, by approximating the given body with a large amount of small cubes or concentric cylindrical shells, and adding the individual volumes of those shapes. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis The volume of irregularly shaped objects can be determined by displacement. In Fluid mechanics, displacement occurs when an object is immersed in a Fluid, pushing it out of the way and taking its 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. In Mathematics, a content is a Real function \mu defined on a Field of sets \mathcal{A} such that \mu(A\in\ In differential geometry, volume is expressed by means of the volume form. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold.

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). The litre or liter (see spelling differences) is a unit of Volume. CM3 redirects here If you were looking for the 3rd game in the Cooking Mama series abbreviated as CM3 see here. 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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

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 to pressure. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Thermodynamics, the Internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

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

1 Volume formulas
2 Volume measures: cooking
3 Relationship to density
4 Volume formula derivation
5 See also
6 External links

Volume formulas

Common equations for volume:
Shape	Equation	Variables
A cube:	$s 3$	s = length of any side
A rectangular prism:	$l \cdot w \cdot h$	l = length, w = width, h = height
A cylinder (circular prism):	$π r 2 h$	r = radius of circular face, h = height
Any prism that has a constant cross sectional area along the height**:	$A \cdot h$	A = 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:	$\frac{1}{3}Ah$	A = 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. In Thermodynamics, the Internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Stress is a measure of the average amount of Force exerted per unit Area. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Thermodynamics and Chemistry, chemical potential, symbolized by μ, is a term introduced by the American engineer chemist and mathematical The particle number, N, is the number of so called ' Elementary particles (or elementary constituents in a thermodynamical system. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an Ellipse. Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space 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, 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. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

Volume measures: cooking

Traditional cooking measures for volume also include:

teaspoon = 1/6 U. A teaspoon is a small Spoon, or a spoon used in measuring commonly used to stir the contents of a cup of Tea or Coffee. S. fluid ounce (about 4. 929 mL)
teaspoon = 1/6 Imperial fluid ounce (about 4. 736 mL)
teaspoon = 5 mL (metric)
tablespoon = ½ U. A tablespoon is a type of Spoon used for serving Measure of volume It is also a measure of Volume used in cooking 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 fluidrams (about 17. The dram (archaic spelling drachm; Apothecary symbol ℨ) was historically both a Coin and a Weight. 76 mL) (British)
cup = 8 U. The cup is a unit of measurement for volume used in cooking to measure bulk foods such as Granulated sugar (dry measurement and liquids ( Fluid measurement 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 density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

The term specific volume is used for volume divided by mass. Specific volume (v is the volume occupied by a unit of mass of a material This is the reciprocal of the mass density, expressed in units such as cubic meters per kilogram (m³·kg^-1). In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

Volume formula derivation

Shape	Volume formula derivation
Sphere	The volume of a sphere is the integral of infinitesimal circular slabs of width $d x$ . "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space 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 of the circular slab is $\pi \cdot y^2$ The volume of the sphere can be calculated as $\int_{-r}^r \pi(r^2-x^2) \,dx$ Replacing $x$ by $x \cdot r$ , so that the integral boundaries become -1 and +1, we get $\pi r^3 \int_{-1}^1 (1-x^2) \,dx$ The antiderivative needed can be determined very easily as $x-\frac{x^3}{3}$ Thus, the sphere volume amounts to V_sphere = $\pi r^3 \cdot[1-1/3-(-1+1/3)]$ = $\frac{4}{3}\pi r^3$ This formula can be derived more quickly using the formula for the sphere surface area, which is $4π r 2$ . In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative Surface area is the measure of how much exposed Area an object has The volume of the sphere consists of layers of infinitesimal spherical slabs, and THE sphere volume is equal to $\int_0^r 4\pi r^2 \,dr$ = $\frac{4}{3}\pi r^3$

Shape

Volume formula derivation

Sphere

The volume of a sphere is the integral of infinitesimal circular slabs of width

d x

. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

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 of the circular slab is $\pi \cdot y^2$
The volume of the sphere can be calculated as $\int_{-r}^r \pi(r^2-x^2) \,dx$
Replacing $x$ by $x \cdot r$ , so that the integral boundaries become -1 and +1, we get $\pi r^3 \int_{-1}^1 (1-x^2) \,dx$
The antiderivative needed can be determined very easily as $x-\frac{x^3}{3}$
Thus, the sphere volume amounts to V_sphere = $\pi r^3 \cdot[1-1/3-(-1+1/3)]$ = $\frac{4}{3}\pi r^3$

This formula can be derived more quickly using the formula for the sphere surface area, which is $4π r 2$ . In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative Surface area is the measure of how much exposed Area an object has The volume of the sphere consists of layers of infinitesimal spherical slabs, and THE sphere volume is equal to

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

External links

Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Conversion of units refers to conversion factors between different Units of measurement for the same Quantity. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different The pages linked in the right-hand column contain lists of volumes that are of the same order of magnitude (power of ten Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Units of mass There are three similar units of Mass called the ton: Long ton (simply ton in countries such as the United In the Physical sciences weight is a Measurement of the gravitational Force acting on an object Dimensioning is the process of measuring the cubic space that a package or object occupies Dimensional weight, used in Shipping and freight, is a Billing technique which takes into account the Volume of a Package.

Dictionary

-noun

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.)
Strength of sound. Measured in decibels. (Please turn down the volume on the stereo.)
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.)
A single book of a publication issued in multi-book format, such as an encyclopedia. (The letter "G" was found in volume 4.)
A synonym for quantity. (The volume of ticket sales decreased this week.)
(economics) The total supply of money in circulation or, less frequently, total amount of credit extended, within a specified national market or worldwide.

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Contents

Volume formulas

Volume measures: cooking

Relationship to density

Volume formula derivation

See also

External links

Dictionary

volume

-noun