From Wikipedia, the free encyclopedia Volume
Volume
In differential geometry, volume is expressed by means
of the volume form, and is an important global Riemann-
ian invariant. In thermodynamics, volume is a fundamen-
tal parameter, and is a conjugate variable to pressure.
Units
Volume measurements from the 1914 The New Student’s Refer-
ence Work.
Approximate conversion to millilitres:[3]
Imperial U.S. liquid U.S. dry
Gill 142 ml 118 ml 138 ml
Pint 568 ml 473 ml 551 ml
A measuring cup can be used to measure volumes of liquids. Quart 1137 ml 946 ml 1101 ml
This cup measures volume in units of cups, fluid ounces, and
millilitres. Gallon 4546 ml 3785 ml 4405 ml
Volume is the quantity of three-dimensional space en- Any unit of length gives a corresponding unit of volume,
closed by some closed boundary, for example, the space namely the volume of a cube whose side has the given
that a substance (solid, liquid, gas, or plasma) or shape length. For example, a cubic centimetre (cm3) would be
occupies or contains.[1] Volume is often quantified nu- the volume of a cube whose sides are one centimetre
merically using the SI derived unit, the cubic metre. The (1 cm) in length.
volume of a container is generally understood to be the In the International System of Units (SI), the standard
capacity of the container, i. e. the amount of fluid (gas unit of volume is the cubic metre (m3). The metric system
or liquid) that the container could hold, rather than the also includes the litre (L) as a unit of volume, where one
amount of space the container itself displaces. litre is the volume of a 10-centimetre cube. Thus
Three dimensional mathematical shapes are also as- 1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001
signed volumes. Volumes of some simple shapes, such as cubic metres,
regular, straight-edged, and circular shapes can be easi-
so
ly calculated using arithmetic formulas. The volumes of
1 cubic metre = 1000 litres.
more complicated shapes can be calculated by integral
calculus if a formula exists for the shape’s boundary. Small amounts of liquid are often measured in millilitres,
One-dimensional figures (such as lines) and two-dimen- where
sional shapes (such as squares) are assigned zero volume 1 millilitre = 0.001 litres = 1 cubic centimetre.
in the three-dimensional space.
The volume of a solid (whether regularly or irregu- Various other traditional units of volume are also in use,
larly shaped) can be determined by fluid displacement. including the cubic inch, the cubic foot, the cubic mile,
Displacement of liquid can also be used to determine the the teaspoon, the tablespoon, the fluid ounce, the fluid
volume of a gas. The combined volume of two substances dram, the gill, the pint, the quart, the gallon, the minim,
is usually greater than the volume of one of the sub- the barrel, the cord, the peck, the bushel, and the
stances. However, sometimes one substance dissolves in hogshead.
the other and the combined volume is not additive.[2]
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From Wikipedia, the free encyclopedia Volume
Shape Volume formula Variables
Cube a = length of any side (or edge)
Cylinder r = radius of circular face, h = height
Prism B = area of the base, h = height
Rectangular l = length, w = width, h = height
prism
Sphere r = radius of sphere
which is the integral of the surface area of a sphere
Ellipsoid a, b, c = semi-axes of ellipsoid
Pyramid B = area of the base, h = height of pyramid
Cone r = radius of circle at base, h = distance from base to tip
Tetrahedron[4] edge length a
Parallelepiped a, b, and c are the parallelepiped edge lengths, and α, β, and γ are
the internal angles between the edges
Any volumet- h = any dimension of the figure,
ric sweep A(h) = area of the cross-sections perpendicular to h described as
(calculus re- a function of the position along h. a and b are the limits of inte-
quired) gration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be
determined from h).
Any rotated RO and RI are functions expressing the outer and inner radii of
figure (washer the function, respectively.
method)
(calculus re-
quired)
Klein bottle No volume—it has no inside.
Related terms ume of a working fluid is often an important parameter
of a system being studied.
Volume and capacity are sometimes distinguished, with The volumetric flow rate in fluid dynamics is the vol-
capacity being used for how much a container can hold ume of fluid which passes through a given surface per
(with contents measured commonly in litres or its de- unit time (for example cubic meters per second [m3 s-1]).
rived units), and volume being how much space an object
displaces (commonly measured in cubic metres or its de-
rived units).
Volume formulas
Volume and capacity are also distinguished in capac-
ity management, where capacity is defined as volume Ratio of volumes of a cone, sphere and
over a specified time period. However in this context the cylinder of the same radius and height
term volume may be more loosely interpreted to mean The above formulas can be used to show that the volumes
quantity. of a cone, sphere and cylinder of the same radius and
The density of an object is defined as mass per unit height are in the ratio 1 : 2 : 3, as follows.
volume. The inverse of density is specific volume which Let the radius be r and the height be h (which is 2r for
is defined as volume divided by mass. Specific volume is the sphere).
a concept important in thermodynamics where the vol-
2
From Wikipedia, the free encyclopedia Volume
Cone
The cone is a type of pyramidal shape. The fundamental
equation for pyramids, one-third times base times alti-
tude, applies cones as well. But for an explanation using
calculus:
The volume of a cone is the integral of infinitesimal
circular slabs of thickness dx. The calculation for the vol-
ume of a cone of height h, whose base is centered at
(0,0,0) with radius r, is as follows.
A cone, sphere and cylinder of radius r and height h 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 discovery of the 2 : 3 ratio of the volumes of the
sphere and cylinder is credited to Archimedes.[5]
The volume of the cone can then be calculated as
Volume formula derivations
Sphere and after extraction of the constants:
The volume of a sphere is the integral of infinitesimal cir-
cular slabs 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 slab is πr2.
The radius of the circular slabs, defined such that the
x-axis cuts perpendicularly through them, is; Integrating gives us
or
Pyramidal shapes
For any shape in three dimensions in which a single base
can be identified and a single peak exists non-planer to
where y or z can be taken to represent the radius of a the base, the pyramidal rule applies.
slab at a particular x value. The volume of any pyramidal shape is one-third of
Using y as the slab radius, the volume of the sphere the product of the base area and the length of the altitude
can be calculated as (the line-segment perpendicular to the plane that the
base is in) of the peak above the plane of the base.
Now
See also
Combining yields gives • Orders of magnitude (volume)
This formula can be derived more quickly using the • Length
formula for the sphere’s surface area, which is 4πr2. The • Perimeter
volume of the sphere consists of layers of infinitesimal • Area
spherical slabs, and the sphere volume is equal to • Mass
• Weight
• Conversion of units
=
• Dimensional weight
• Dimensioning
• Volume form
• Volume (thermodynamics)
3
From Wikipedia, the free encyclopedia Volume
• Banach–Tarski paradox [3] "General Tables of Units of Measurement". NIST
Weights and Measures Division. http://ts.nist.gov/
References WeightsAndMeasures/Publications/appxc.cfm#4e.
Retrieved 2011-01-12.
[1] "Your Dictionary entry for "volume"". [4] Coxeter, H. S. M.: Regular Polytopes (Methuen and
http://www.yourdictionary.com/volume. Co., 1948). Table I(i).
Retrieved 2010-05-01. [5] Rorres, Chris. "Tomb of Archimedes: Sources".
[2] One litre of sugar (about 970 grams) can dissolve in Courant Institute of Mathematical Sciences.
0.6 litres of hot water, producing a total volume of http://www.math.nyu.edu/~crorres/Archimedes/
less than one litre. "Solubility". Tomb/Cicero.html. Retrieved 2007-01-02.
http://chemed.chem.purdue.edu/genchem/
topicreview/bp/ch18/soluble.php. Retrieved
2010-05-01. "Up to 1800 grams of sucrose can
External links
dissolve in a liter of water." • Volume calculator - Javascript automatic calculator.
Retrieved from "http://en.wikipedia.org/w/index.php?title=Volume&oldid=462962411"
Categories:
• Fundamental physics concepts
• Volume
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