Volume Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume Volume is often quantified numerically using the SI derived unit, the cubic metre. 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. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape's boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using three-dimensional techniques such as the Body Volume Volume Index. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) 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. 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. In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.Contents4.1 Volume Volume ratios for a cone, sphere and cylinder of the same radius and height 4.2 Volume Volume formula derivations4.2.1 Sphere 4.2.2 Cone 4.2.3 Polyhedron5 Volume Volume in differential geometry 6 Volume Volume in thermodynamics 7 See also 8 References 9 External linksUnits Volume Volume measurements from the 1914 The New Student's Reference Work.Approximate conversion to metric (mL)Imp. U.S.Liquid DryGill 142 118 138Pint 568 473 551Quart 1137 946 1101Gallon 4546 3785 4405Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm3) is the volume of a cube whose sides are one centimetre (1 cm) in length. In the International System of Units International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,so1 cubic metre = 1000 litres.Small amounts of liquid are often measured in millilitres, where1 millilitre = 0.001 litres = 1 cubic centimetre.Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, and the hogshead. Related terms Capacity is defined by the Oxford English Dictionary Oxford English Dictionary as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them". (The word capacity has other unrelated meanings, as in e.g. capacity management.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle's fuel tank is rarely stated in cubic feet, for example, but in gallons (a gallon fills a volume of 0.1605 cu ft). The density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied. The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s−1]). Volume Volume in calculus Further information: Volume Volume element In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f ( x , y , z ) = 1 displaystyle f(x,y,z)=1 and is usually written as: ∭ D 1 d x d y d z . displaystyle iiint limits _ D 1,dx,dy,dz. The volume integral in cylindrical coordinates is ∭ D r d r d θ d z , displaystyle iiint limits _ D r,dr,dtheta ,dz, and the volume integral in spherical coordinates (using the convention for angles with θ displaystyle theta as the azimuth and ϕ displaystyle phi measured from the polar axis; see more on conventions) has the form ∭ D ρ 2 sin ⁡ ϕ d ρ d θ d ϕ . displaystyle iiint limits _ D rho ^ 2 sin phi ,drho ,dtheta ,dphi . Cube a 3 displaystyle a^ 3 ; a = length of any side (or edge)Circular Cylinder π r 2 h displaystyle pi r^ 2 h; r = radius of circular base, h = heightPrism B h displaystyle Bh B = area of the base, h = heightCuboid l w h displaystyle lwh l = length, w = width, h = heightTriangular prism 1 2 b h l displaystyle frac 1 2 bhl b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases 1 4 h − a 4 + 2 ( a b ) 2 + 2 ( a c ) 2 − b 4 + 2 ( b c ) 2 − c 4 displaystyle frac 1 4 h sqrt -a^ 4 +2(ab)^ 2 +2(ac)^ 2 -b^ 4 +2(bc)^ 2 -c^ 4 a, b, and c = lengths of sides h = height of the triangular prismSphere 4 3 π r 3 = 1 6 π d 3 displaystyle frac 4 3 pi r^ 3 = frac 1 6 pi d^ 3 r = radius of sphere d = diameter of sphere which is the integral of the surface area of a sphereEllipsoid 4 3 π a b c displaystyle frac 4 3 pi abc a, b, c = semi-axes of ellipsoidTorus ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 displaystyle left(pi r^ 2 right)left(2pi Rright)=2pi ^ 2 Rr^ 2 r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)Pyramid 1 3 B h displaystyle frac 1 3 Bh B = area of the base, h = height of pyramidSquare pyramid 1 3 s 2 h displaystyle frac 1 3 s^ 2 h; s = side length of base, h = height 1 3 l w h displaystyle frac 1 3 lwh l = length, w = width, h = heightCone 1 3 π r 2 h displaystyle frac 1 3 pi r^ 2 h r = radius of circle at base, h = distance from base to tip or heightRegular tetrahedron 2 12 a 3 displaystyle sqrt 2 over 12 a^ 3 , Edge length, a displaystyle a Parallelepiped a b c K displaystyle abc sqrt K K = 1 + 2 cos ⁡ ( α ) cos ⁡ ( β ) cos ⁡ ( γ ) − cos 2 ⁡ ( α ) − cos 2 ⁡ ( β ) − cos 2 ⁡ ( γ ) displaystyle begin aligned K=1&+2cos(alpha )cos(beta )cos(gamma )\&-cos ^ 2 (alpha )-cos ^ 2 (beta )-cos ^ 2 (gamma )end aligned a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edgesAny volumetric sweep (calculus required) ∫ a b A ( h ) d h displaystyle int _ a ^ b A(h),mathrm d h 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. a and b are the limits of integration for the volumetric sweep.(This will work for any figure if its cross-sectional area can be determined from h).Any rotated figure (washer method; calculus required) π ∫ a b ( [ R O ( x ) ] 2 − [ R I ( x ) ] 2 ) d x displaystyle pi int _ a ^ b left( left[R_ O (x)right] ^ 2 - left[R_ I (x)right] ^ 2 right)mathrm d x R O displaystyle R_ O and R I displaystyle R_ I are functions expressing the outer and inner radii of the function, respectively. Volume Volume ratios for a cone, sphere and cylinder of the same radius and heightA cone, sphere and cylinder of radius r and height hThe above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows. Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is 1 3 π r 2 h = 1 3 π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 1 , displaystyle frac 1 3 pi r^ 2 h= frac 1 3 pi r^ 2 left(2rright)=left( frac 2 3 pi r^ 3 right)times 1, the volume of the sphere is 4 3 π r 3 = ( 2 3 π r 3 ) × 2 , displaystyle frac 4 3 pi r^ 3 =left( frac 2 3 pi r^ 3 right)times 2, while the volume of the cylinder is π r 2 h = π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 3. displaystyle pi r^ 2 h=pi r^ 2 (2r)=left( frac 2 3 pi r^ 3 right)times 3. The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes. Volume Volume formula derivations Sphere The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows. The surface area of the circular disk is π r 2 displaystyle pi r^ 2 . The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is y = r 2 − x 2 displaystyle y= sqrt r^ 2 -x^ 2 or z = r 2 − x 2 displaystyle z= sqrt r^ 2 -x^ 2 where y or z can be taken to represent the radius of a disk at a particular x value. Using y as the disk radius, the volume of the sphere can be calculated as ∫ − r r π y 2 d x = ∫ − r r π ( r 2 − x 2 ) d x . displaystyle int _ -r ^ r pi y^ 2 ,dx=int _ -r ^ r pi left(r^ 2 -x^ 2 right),dx. Now ∫ − r r π r 2 d x − ∫ − r r π x 2 d x = π ( r 3 + r 3 ) − π 3 ( r 3 + r 3 ) = 2 π r 3 − 2 π r 3 3 . displaystyle int _ -r ^ r pi r^ 2 ,dx-int _ -r ^ r pi x^ 2 ,dx=pi left(r^ 3 +r^ 3 right)- frac pi 3 left(r^ 3 +r^ 3 right)=2pi r^ 3 - frac 2pi r^ 3 3 . Combining yields V = 4 3 π r 3 . displaystyle V= frac 4 3 pi r^ 3 . This formula can be derived more quickly using the formula for the sphere's surface area, which is 4 π r 2 displaystyle 4pi r^ 2 . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to ∫ 0 r 4 π r 2 d r = 4 3 π r 3 . displaystyle int _ 0 ^ r 4pi r^ 2 ,dr= frac 4 3 pi r^ 3 . Cone The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well. However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0, 0, 0) with radius r, is as follows. The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is, r h − x h . displaystyle r frac h-x h . The surface area of the circular disk is then π ( r h − x h ) 2 = π r 2 ( h − x ) 2 h 2 . displaystyle pi left(r frac h-x h right)^ 2 =pi r^ 2 frac (h-x)^ 2 h^ 2 . The volume of the cone can then be calculated as ∫ 0 h π r 2 ( h − x ) 2 h 2 d x , displaystyle int _ 0 ^ h pi r^ 2 frac (h-x)^ 2 h^ 2 dx, and after extraction of the constants π r 2 h 2 ∫ 0 h ( h − x ) 2 d x displaystyle frac pi r^ 2 h^ 2 int _ 0 ^ h (h-x)^ 2 dx Integrating gives us π r 2 h 2 ( h 3 3 ) = 1 3 π r 2 h . displaystyle frac pi r^ 2 h^ 2 left( frac h^ 3 3 right)= frac 1 3 pi r^ 2 h. Polyhedron Main article: Volume Volume of a polyhedron Volume Volume in differential geometry Main article: Volume Volume form In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. Integrating the volume form gives the volume of the manifold according to that form. An oriented pseudo-Riemannian manifold has a natural volume form. In local coordinates, it can be expressed as ω = g d x 1 ∧ ⋯ ∧ d x n , displaystyle omega = sqrt g ,dx^ 1 wedge dots wedge dx^ n , where the d x i displaystyle dx^ i are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold, and g displaystyle g is the determinant of the matrix representation of the metric tensor on the manifold in terms of the same basis. Volume Volume in thermodynamics Main article: Volume Volume (thermodynamics) In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law. See alsoArea Banach–Tarski paradox Conversion of units Dimensional weight Dimensioning Length Measure Orders of magnitude (volume) Perimeter Volume Volume (thermodynamics) Volumography WeightReferences^ "Your Dictionary entry for "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". Retrieved 2010-05-01. Up to 1800 grams of sucrose can dissolve in a liter of water.  ^ "General Tables of Units of Measurement". NIST Weights and Measures Division. Archived from the original on 2011-12-10. Retrieved 2011-01-12.  ^ "capacity". Oxford English Dictionary Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005.  (Subscription or UK public library membership required.) ^ "density". Oxford English Dictionary Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005.  (Subscription or UK public library membership required.) ^ Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i). ^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02. External linksWikimedia Commons has media related to Volume.

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