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=== ఒకే వ్యాసార్థం, ఎత్తు గల శంకువు, గోళం, స్థూపం ఘనపరిమాణాల నిషత్తులు ===
=== Volume ratios for a cone, sphere and cylinder of the same radius and height ===
[[File:Inscribed_cone_sphere_cylinder.svg|link=https://en.wikipedia.org/wiki/File:Inscribed_cone_sphere_cylinder.svg|thumb|350x350px|A cone, sphere and cylinder of radius ''r'' and height ''h'']]
పైన సూచించిన సూత్రములు ఉపయోగించి ఒకే వ్యాసార్థం, ఎత్తు గల శంకువు, గోళం, స్థూపం ఘనపరిమాణాలను గణన చేస్తే వాతి ఘనపరిమానముల నిష్పత్తి '''1 : 2 : 3''', ఉంటుంది.
The above formulas can be used to show that the volumes of a [[:en:Cone_(geometry)|cone]], sphere and [[:en:Cylinder_(geometry)|cylinder]] of the same radius and height are in the ratio '''1 : 2 : 3''', as follows.
వ్యాసార్థం ''r'' , ఎత్తు ''h'' ( 2''r'' ),అయినపుడు శంకువు ఘనపరిమాణం
Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere), then the volume of the cone is
: <math>\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 \left(2r\right) = \left(\frac{2}{3} \pi r^3\right) \times 1,</math>
గోళం ఘనపరిమాణం :
the volume of the sphere is
: <math>\frac{4}{3} \pi r^3 = \left(\frac{2}{3} \pi r^3\right) \times 2,</math>
స్థూపం ఘనపరిమాణం:
while the volume of the cylinder is
: <math>\pi r^2 h = \pi r^2 (2r) = \left(\frac{2}{3} \pi r^3\right) \times 3.</math>
Theగోళం, discoveryస్థూపం ofల theఘనపరిమాణాల నిష్పత్తి '''2 : 3''' ratio ofఅని theఆర్కిమెడిస్ volumesఅనే ofశాస్త్రవేత్త theకనుగొన్నాడు. sphere and cylinder is credited to [[:en:Archimedes|Archimedes]].<ref>{{cite web|url=http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title=Tomb of Archimedes: Sources|last=Rorres|first=Chris|publisher=Courant Institute of Mathematical Sciences|access-date=2007-01-02}}</ref>
=== Volume formula derivations ===
==== Polyhedron ====
{{main|Volume of a polyhedron}}
== Volume in differential geometry ==
{{main|Volume form}}In [[:en:Differential_geometry|differential geometry]], a branch of [[:en:Mathematics|mathematics]], a '''volume form''' on a [[:en:Differentiable_manifold|differentiable manifold]] is a [[:en:Differential_form|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 [[:en:Density_on_a_manifold|density]]. Integrating the volume form gives the volume of the manifold according to that form.
An [[:en:Orientation_(mathematics)|oriented]] [[:en:Pseudo-Riemannian_manifold|pseudo-Riemannian manifold]] has a natural volume form. In [[:en:Local_coordinates|local coordinates]], it can be expressed as
== Volume in thermodynamics ==
{{Main|Volume (thermodynamics)}}In [[:en:Thermodynamics|thermodynamics]], the '''volume''' of a [[:en:Thermodynamic_system|system]] is an important [[:en:Extensive_parameter|extensive parameter]] for describing its [[:en:Thermodynamic_state|thermodynamic state]]. The '''specific volume''', an [[:en:Intensive_property|intensive property]], is the system's volume per unit of mass. Volume is a [[:en:Function_of_state|function of state]] and is interdependent with other thermodynamic properties such as [[:en:Pressure|pressure]] and [[:en:Thermodynamic_temperature|temperature]]. For example, volume is related to the [[:en:Pressure|pressure]] and [[:en:Thermodynamic_temperature|temperature]] of an [[:en:Ideal_gas|ideal gas]] by the [[:en:Ideal_gas_law|ideal gas law]].
== Volume computation ==
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