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| Units | |
| 2>
Volume measurements from the 1914 The New Student's Reference 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
Quart
1137 ml
946 ml
1101 ml
Gallon
4546 ml
3785 ml
4405 ml
Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre (cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length.
In the 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. Thus
1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,
so
1 cubic metre = 1000 litres.
Small amounts of liquid are often measured in millilitres, where
1 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.
[edit] Tags:Liquid,The New Student's Reference Work,Gill,Pint,Quart,Gallon,Cube,Metric System,Cubic Inch,Cubic Foot,Cubic Mile,Teaspoon,Tablespoon,Fluid Ounce,Fluid Dram,Minim,Barrel,Cord,Peck,Bushel,Hogshead,Length,Measure, | |
| Related terms | |
| 2>
Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units).
Volume and capacity are also distinguished in capacity management, where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity.
The density of an object is defined as mass per unit 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]).
[edit] Tags:Quantity,Space,Thermodynamics,Mass,Contents, | |
| Volume formulas | |
| 2>
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 prism
l = length, w = width, h = height
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 volumetric sweep
(calculus 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. 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)
RO and RI are functions expressing the outer and inner radii of the function, respectively.
Klein bottle
No volume—it has no inside.
[edit] Tags:Integral,Surface Area,Circle,Tetrahedron,Cone,Sphere,Area, | |
| Ratio of volumes of a cone, sphere and cylinder of the same radius and height | |
| 3>
A cone, sphere and cylinder of radius r and height h
The 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
the volume of the sphere is
while the volume of the cylinder is
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[5]
[edit] Tags:Archimedes, | |
| Sphere | |
| 3>
The volume of a sphere is the 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 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;
or
where y or z can be taken to represent the radius of a slab at a particular x value.
Using y as the slab radius, the volume of the sphere can be calculated as
Now
Combining yields gives
This formula can be derived more quickly using the formula for the sphere's surface area, which is 4πr2. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to
=
[edit] Tags: | |
| Cone | |
| 3>
The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, 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 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 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
[edit] Tags: | |
| See also | |
| 2>
Orders of magnitude (volume)
Length
Perimeter
Area
Measure
Mass
Weight
Conversion of units
Dimensional weight
Dimensioning
Volume form
Volume (thermodynamics)
Banach–Tarski paradox
[edit] Tags:Perimeter,Weight,Dimensional Weight,Dimensioning,Banach–tarski Paradox, | |
| References | |
| 2>
^ "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."
^ "General Tables of Units of Measurement". NIST Weights and Measures Division. http://ts.nist.gov/WeightsAndMeasures/Publications/appxc.cfm#4e. Retrieved 2011-01-12.
^ Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).
^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html. Retrieved 2007-01-02.
[edit] Tags:Coxeter, H. S. M.,Regular Polytopes, | |
| External links | |
| 2>
The Wikibook Geometry has a page on the topic of
Perimeters, Areas, Volumes
The Wikibook Calculus has a page on the topic of
Volume
Volume calculator - Javascript automatic calculator.
Retrieved from "http://en.wikipedia.org/w/index.php?title=Volume&oldid=476111477"
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