Variance
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![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Simple_Measuring_Cup.jpg/300px-Simple_Measuring_Cup.jpg)
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.[1] 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. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. 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.[2]
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.
Units
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Volume_measurements_from_The_New_Student%27s_Reference_Work.svg/220px-Volume_measurements_from_The_New_Student%27s_Reference_Work.svg.png)
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.
Related terms
Volume and capacity are the 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]).
Volume formulas
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 | |
Triangular prism | b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases | |
Sphere | r = radius of sphere which is the integral of the surface area of a sphere | |
Ellipsoid | a, b, c = semi-axes of ellipsoid | |
Torus | r = minor radius, R = major radius | |
Pyramid | B = area of the base, h = height of pyramid | |
Square pyramid | s = side length of base, h = height | |
Rectangular pyramid | l = length, w = width, h = height | |
Cone | r = radius of circle at base, h = distance from base to tip or height | |
Tetrahedron[4] | edge length | |
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) |
and are functions expressing the outer and inner radii of the function, respectively. |
Volume ratios for a cone, sphere and cylinder of the same radius and height
![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Inscribed_cone_sphere_cylinder.svg/350px-Inscribed_cone_sphere_cylinder.svg.png)
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]
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 .
The radius of the circular disks, 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 disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
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 small 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,
The surface area of the circular disk is then
The volume of the cone can then be calculated as
and after extraction of the constants:
See also
References
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External links
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Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014
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Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT
- Volume calculator – Javascript automatic calculator.
- ↑ Template:Cite web
- ↑ 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. Template:Cite web
- ↑ Template:Cite web
- ↑ Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).
- ↑ Template:Cite web