The volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. 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. Volume is commonly presented in units such as cubic meters, cubic centimeters, litres, or millilitres.

Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas In mathematics and other sciences, a formula (plural: formulas or formulae is a concise way of expressing information symbolically , or a general relationship between quantities. One of many famous formulae is Albert Einstein's E = mc2 (see special relativity). More complicated shapes can be calculated by integral calculus Integration is an important concept in mathematics which, together with differentiation, forms one of the 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 its boundary. The volume of any shape 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 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.

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.

Volume is a fundamental parameter in thermodynamics In physics, thermodynamics is the study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature,volume and pressure. Its underpinnings, based upon statistical predictions of the collective motion of particles from their microscopic behavior, is the field of statistical thermodynamics (or and it is conjugate 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.

Conjugate variables of thermodynamics
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 Volume
(Stress In continuum mechanics, the concept of stress, introduced by Cauchy around 1822, is a measure of the average amount of force exerted per unit area of a surface within a deformable body on which internal forces act . In other words, it is a measure of the intensity or internal distribution of the total internal forces acting within a deformable) (Strain In continuum mechanics, deformation is the change in shape and/or size of a continuum body after it undergoes a displacement between an initial or undeformed configuration , at time , and a current or deformed configuration , at the current time .[clarification needed])
Temperature In physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics. If no net heat flow occurs between two objects, the objects have the same temperature; otherwise heat flows Entropy Entropy is a concept of information maintaining great importance in physics, chemistry, and information theory . When given a system whose exact description is unknown, its entropy is defined as the amount of information needed to exactly specify the state of the system (to the full extent that it can be described in the universe itself). This is
Chem. potential Chemical potential, symbolized by μ, is a quantity first described by the American engineer, chemist and mathematical physicist Josiah Williard Gibbs. He defined it as follows: Particle no. The particle number, N, is the number of constituent particles in a thermodynamical system. The particle number is a fundamental parameter in thermodynamics and it is conjugate to the chemical potential

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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 of an object is defined as mass per unit volume. The inverse of density is specific volume where, R is the specific gas constant, M is the molar mass, T is the temperature and P is the pressure of the gas 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 liters The litre or liter 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 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

measure US Imperial metric
teaspoon A teaspoon, a type of cutlery , is a small spoon, commonly silver and part of a 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., tblspn, tbs. or tbsp. Canada, Japan, New Zealand, South Africa and the UK define 1 level tablespoon as 15 ml = 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 unit of measurement for volume, used in cooking to measure liquids and bulk foods such as granulated sugar (dry measurement) in many countries, although different countries use different sizes. 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 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 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).

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 which, together with differentiation, forms one of the 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 then 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 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, round 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" sometimes refers (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 the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius at base, h = distance from base to tip
Any figure (calculus Integration is an important concept in mathematics which, together with differentiation, forms one of the 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. It is to a parallelogram as a cube is to a square: Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds. Three equivalent definitions of parallelepiped are.

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 which, together with differentiation, forms one of the 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 πy2.

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 then 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, round 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" sometimes refers is the integral Integration is an important concept in mathematics which, together with differentiation, forms one of the 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 , 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 Wikibooks is a Wiki hosted by the Wikimedia Foundation for the creation of free content textbooks and annotated texts that anyone can edit Calculus has a page on the topic of Volume
The Wikibook Wikibooks is a Wiki hosted by the Wikimedia Foundation for the creation of free content textbooks and annotated texts that anyone can edit Geometry has a page on the topic of Perimeters, Areas, Volumes

External links

Categories: Fundamental physics concepts | Volume

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