Editing Volume (section)

== Computation ==

=== Basic shapes ===
[[File:visual_proof_cone_volume.svg|thumb|[[Proof without words]] that the volume of a cone is a third of a cylinder of equal diameter and height
{|
|valign="top"|{{nowrap|1.}}||A cone and a cylinder have {{nowrap|radius ''r''}} and {{nowrap|height ''h''.}}
|-
|valign="top"|2.||The volume ratio is maintained when the height is scaled to {{nowrap|1=''h' ''= ''r'' &#8730;{{pi}}.}}
|-
|valign="top"|3.||Decompose it into thin slices.
|-
|valign="top"|4.||Using Cavalieri's principle, reshape each slice into a square of the same area.
|-
|valign="top"|5.||The pyramid is replicated twice.
|-
|valign="top"|6.||Combining them into a cube shows that the volume ratio is 1:3.
|}]]
{{See also|List of formulas in elementary geometry}}For many shapes such as the [[cube]], [[cuboid]] and [[cylinder]], they have an essentially the same volume calculation formula as one for the [[Prism (geometry)|prism]]: the [[Base (geometry)|base]] of the shape multiplied by its [[height]].

=== Integral calculus ===
{{Further|Volume integral}}[[File:Integral_apl_rot_objem3.svg|alt=f(x) and g(x) rotated in the x-axis|thumb|Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x).]]
The calculation of volume is a vital part of [[integral]] calculus. One of which is calculating the volume of [[Solid of revolution|solids of revolution]], by rotating a [[plane curve]] around a [[Line (geometry)|line]] on the same plane. The washer or [[disc integration]] method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:<math display="block">V = \pi \int_a^b \left| f(x)^2 - g(x)^2\right|\,dx</math>where <math display="inline">f(x)</math> and <math display="inline">g(x)</math> are the plane curve boundaries.<ref name="RIT-2014">{{Cite web |date=22 September 2014 |title=Volumes by Integration |url=https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/C8_VolumesbyIntegration_BP_9_22_14.pdf |url-status=live |archive-url=https://web.archive.org/web/20220202194113/https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/C8_VolumesbyIntegration_BP_9_22_14.pdf |archive-date=2 February 2022 |access-date=12 August 2022 |website=[[Rochester Institute of Technology]] }}</ref>{{Rp|pages=1,3}} The [[shell integration]] method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as:<ref name="RIT-2014"/>{{Rp|pages=6}}<math display="block">V = 2\pi \int_a^b x |f(x) - g(x)|\, dx</math> The volume of a [[region (mathematics)|region]] ''D'' in [[three-dimensional space]] is given by the triple or [[volume integral]] of the constant [[function (mathematics)|function]] <math>f(x,y,z) = 1</math> over the region. It is usually written as:<ref>{{cite book |last=Stewart |first=James |url=https://archive.org/details/calculusearlytra00stew_1 |title=Calculus: Early Transcendentals |date=2008 |publisher=Brooks Cole Cengage Learning |isbn=978-0-495-01166-8 |edition=6th |author-link=James Stewart (mathematician) |url-access=registration}}</ref>{{rp|at=Section 14.4}}
<math display="block">\iiint_D 1 \,dx\,dy\,dz.</math>

In [[cylindrical coordinate system|cylindrical coordinates]], the [[volume integral]] is
<math display="block">\iiint_D r\,dr\,d\theta\,dz, </math>

In [[spherical coordinate system|spherical coordinates]] (using the convention for angles with <math>\theta</math> as the azimuth and <math>\varphi</math> measured from the polar axis; see more on [[Spherical coordinate system#Conventions|conventions]]), the volume integral is
<math display="block">\iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .</math>

=== Geometric modeling ===
[[File:Dolphin_triangle_mesh.png|alt=Tiled triangles to form a dolphin shape|thumb|250x250px|[[Low poly]] triangle mesh of a dolphin]]
A [[polygon mesh]] is a representation of the object's surface, using [[polygon]]s. The [[volume mesh]] explicitly define its volume and surface properties.