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 International System of Units specifies a set of seven base units from which all other units of measurement are formed. These other units are called SI derived units and are also considered part of the standard, the cubic metre The cubic metre is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère. Another alternative name, not widely used any more, is the kilolitre. 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 In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language. The volumes of more complicated shapes can be calculated by integral calculus Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral if a formula exists for the shape's boundary. One-dimensional figures (such as lines In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some) and two-dimensional shapes (such as squares In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles . A square with vertices ABCD would be denoted ABCD) 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 In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place. The volume of the fluid displaced can then be measured, as in the illustration, and from this the volume of the immersed object can be deduced. 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 for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation.[2]
In differential geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and, volume is expressed by means of the volume form In mathematics, a volume form on a differentiable manifold is a nowhere vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn(T∗M), that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable, and orientable manifolds have, and is an important global Riemannian Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area, and volume. From those invariant The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive regardless of how we count the objects in the set. The quantity – a cardinal number – is associated with the set and is invariant under the process of. In thermodynamics In science, thermodynamics is the study of energy conversion between heat and mechanical work, and subsequently the macroscopic variables such as temperature, volume and pressure, volume is a fundamental parameter The volume of gas increases proportionally to absolute temperature and decreases inversely proportionally to pressure, approximately according to the ideal gas law:, and is a conjugate variable In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials are expressed in terms of conjugate pairs to pressure Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
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Related terms
The density The density of a material is defined as its mass per unit volume. The symbol of density is ρ . In some countries (for instance, in the United States), density is also defined as its weight per unit volume . The density of a substance is the reciprocal of its specific volume, a representation commonly used in thermodynamics of an object is defined as mass per unit volume. The inverse of density is specific volume Specific volume is the volume occupied by a unit of mass of a material. The specific volume of a substance is equal to the reciprocal of its mass density. Specific volume may be expressed in , or 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 The litre is a unit of volume. There are two official symbols: the Latin letter L in lower and upper case (l and L). The lower case L is also often written as a cursive ℓ, though this symbol has no official approval by any international bureau. Although the litre is not an SI unit, it is accepted for use with the SI, and has appeared in several or its derived units), and volume being how much space an object displaces (commonly measured in cubic meters The cubic metre is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère. Another alternative name, not widely used any more, is the kilolitre or its derived units).
Volume and capacity are also distinguished in a capacity management Capacity Management is a process used to manage information technology . Its primary goal is to ensure that IT capacity meets current and future business requirements in a cost-effective manner. One common interpretation of Capacity Management is described in the ITIL framework . ITIL version 3 views capacity management as comprising three sub- setting, where capacity is defined as volume over a specified time period.
Traditional cooking measures
Further information: United States customary units#Units of capacity and volume The United States customary system is the most commonly used system of measurement in the United States. It is similar but not identical to the British Imperial units. The U.S. is the only industrialized nation that does not mainly use the metric system in its commercial and standards activities, although the International System of Units (SI, and Imperial units#Volume Imperial units or the imperial system is a system of units, first defined in the British Weights and Measures Act of 1824, later refined and reduced. The system came into official use across the British Empire. By the late 20th century most nations of the former empire had officially adopted the metric system as their main system of measurementIn the UK, a tablespoon can also be five fluidrams The dram was historically both a coin and a weight. Currently it is both a small mass in the Apothecaries' system of weights and a small unit of volume. This unit is called more correctly fluid dram or in contraction also fluidram (about 17.76 mL). In Australia, a tablespoon is 4 teaspoons (20 mL).
Volume formulas
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 In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. The tetrahedron is the only convex polyhedron that has four faces and parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just like a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts (i.e., parallelepiped, parallelogram, cube, and square). In this context of affine geometry, in which angles.
Volume formula derivation
Sphere
The volume of a sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The is the integral Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite 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 πr2.
The volume of the sphere can be calculated as
Now and
Combining yields
This formula can be derived more quickly using the formula for the sphere's surface area Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface, which is 4πr2. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to
Cone
The volume of a cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" is the integral Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite 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,
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
See also
| The Wikibook Calculus has a page on the topic of Volume |
| The Wikibook Geometry has a page on the topic of Perimeters, Areas, Volumes |
- Area Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of
- Conversion of units Conversion of units refers to conversion factors between different units of measurement for the same quantity
- Density The density of a material is defined as its mass per unit volume. The symbol of density is ρ . In some countries (for instance, in the United States), density is also defined as its weight per unit volume . The density of a substance is the reciprocal of its specific volume, a representation commonly used in thermodynamics
- Dimensional weight Dimensional weight, used in shipping and freight, is a billing technique which takes into account the XYZ axis dimensions of a package
- Dimensioning
- Length
- Mass
- Orders of magnitude (volume)
- Specific volume
- Volume form
- Weight
- Gas volume
References
- ^ "Your Dictionary entry for "volume"". http://www.yourdictionary.com/volume. Retrieved 2010-05-01.
- ^ 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". http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch18/soluble.php. Retrieved 2010-05-01. "Up to 1800 grams of sucrose can dissolve in a liter of water."
External links
- Volume calculator - Javascript automatic calculator.
Categories: Fundamental physics concepts | Volume
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