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.

Contents

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 measurement
measure US Imperial metric
teaspoon A teaspoon, an item of cutlery , is a small spoon, commonly part of a silverware (usually silver plated, German silver or now, stainless steel) place setting, suitable for stirring and sipping the contents of a cup of tea or coffee. Utilitarian versions are used for measuring 1/6 U.S. fluid ounce (about 4.929 mL) 1/6 Imperial fluid ounce (about 4.736 mL) 5 mL
tablespoon A tablespoon is a type of large spoon usually used for serving. A tablespoonful, an amount approximately equal to the capacity of one tablespoon, is commonly used as a measure of volume in cooking. It is abbreviated in English as T, tb, tbs, tbsp, tblsp, or tblspn. Only the tbs and tbsp abbreviations are currently formally recognized, although the = 3 teaspoons ½ U.S. fluid ounce (about 14.79 mL) ½ Imperial fluid ounce (about 14.21 mL) 15 mL
cup The cup is a customary unit of measurement mainly used in North America for volume, used in cooking to measure liquids and bulk foods such as granulated sugar (dry measurement). This measure is usually used as an informal unit in cooking recipes rather than as a measure for the sale of foodstuffs; precision is rarely required 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 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

Shape Equation Variables
A cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the a = length of any side (or edge)
A rectangular prism: l = length, w = width, h = height
A cylinder 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 of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since: r = radius of circular face, h = height
A general prism In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids: B = area of the base, h = height
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: r = radius of sphere which 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 the 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 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
An ellipsoid An ellipsoid is a closed type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is: a, b, c = semi-axes of ellipsoid
A pyramid In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base: B = area of the base, h = height of pyramid
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" (circular-based pyramid): r = radius of circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre . The common distance of the points of a circle from its centre is called its radius at base, h = distance from base to tip
Any figure (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 required) 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).

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

References

  1. ^ "Your Dictionary entry for "volume"". http://www.yourdictionary.com/volume. Retrieved 2010-05-01.
  2. ^ 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

Categories: Fundamental physics concepts | Volume

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Sat Sep 4 09:05:01 2010
How do I teach the concept of volume to my fifth grade daughter?
Q. For some reason, my very bright daughter is stuck on the concept of VOLUME. I have explained to her that volume is the amount of matter that an object can contain. I used the example of measure cups and said "the volume of a one cup measuring cup is one cup." Anyone have any better ideas? Any links to any video clips online that I could show her?
Asked by RebeccaGirl2007 - Thu Dec 11 07:59:57 2008 - - 7 Answers - 0 Comments

A. Do an at home experiment of displacement. Get a baking dish (something large and deep), put a pitcher full of water all the way to the top in the middle of the baking dish. Then get something like an orange and drop it slowly into the water and let the water over flow into the pan. remove the pitcher with orange in it and pour the water that poured out into a measuring cup and there is your volume of that orange. Then enjoy the orange. Try it with several fruits and veggies to show the volume will differ from every item. Also do it by the sink in case of some spillage.
Answered by dodgeramcharger84 - Thu Dec 11 08:08:11 2008

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