WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

Volume
✖
Volume

✖
Volume[reg]

gives the volume of the three-dimensional region reg.

✖
Volume[{x1,…,xn},{s,smin,smax},{t,tmin,tmax},{u,umin,umax}]

gives the volume of the parametrized region whose Cartesian coordinates xi are functions of s, t, u.

✖
Volume[{x1,…,xn},{s,smin,smax},{t,tmin,tmax},{u,umin,umax},chart]

interprets the xi as coordinates in the specified coordinate chart.

Details and Options

  • A three-dimensional region can be embedded in any dimension greater than or equal to three.
  • In Volume[x,{s,smin,smax},{t,tmin,tmax},{u,umin,umax}], if x is a scalar, Volume returns the volume of the parametric three-region {s,t,u,x}.
  • Coordinate charts in the fifth argument of Volume can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • The following options can be given:
  • AccuracyGoalInfinitydigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecision Automaticthe precision used in internal computations
  • Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

The volume of a unit ball in 3D:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-sy8lbf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-2uvyd5
Out[2]=2

The volume of a standard simplex in 3D:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-cn6j8h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-uzivxt
Out[2]=2

The volume of a rectangular cuboid:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-8wv1d8
Out[1]=1

Volume of the cylinder , expressed in cylindrical coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-8oa2el
Out[1]=1

Scope  (20)Survey of the scope of standard use cases

Special Regions  (7)

Volume of a Cuboid:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-cftuk2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-llsaa6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0tz9x816-3bhqsq
Out[4]=4

Parallelepiped:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-b8zdur
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-0j9llh
Out[2]=2

Simplex in 3D:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-rz305
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-8h00rg
Out[2]=2

A volume simplex embedded in 4D:

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-baxmuk
Out[3]=3

Ball:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-d49pfu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-b7xnwr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-70dknw
Out[3]=3

Ellipsoid:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-y9ip8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-8eal6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-c6qckc
Out[3]=3

Cylinder:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-fd9pjv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-nlo5u
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-fxcmr8
Out[3]=3

Cone:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-i4z5mu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-5k72s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-tqnt0e
Out[3]=3

Formula Regions  (2)

The volume of a ball represented as an ImplicitRegion:

In[4]:=4
✖
https://wolfram.com/xid/0tz9x816-oykaz1
Out[4]=4

A cylinder:

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-cqp0xt
Out[3]=3

The volume of a ball represented as a ParametricRegion:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-emmtka
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-fvitc6
Out[2]=2

A cylinder represented with a rational parametrization:

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-bxbob8
Out[3]=3

Mesh Regions  (2)

The volume of a MeshRegion:

In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-hptxy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-evn5ul
Out[3]=3

The volume of a BoundaryMeshRegion:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-o9vxn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-c4mkmt
Out[2]=2

Derived Regions  (3)

The volume of a RegionIntersection:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-8brni
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-5fe7j0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-1d7a5
Out[3]=3

The volume of a TransformedRegion:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-ebl7ww
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-7hop3r
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-n1impd
Out[3]=3

The volume of a RegionBoundary:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-hkslw5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-ljvzcw
Out[2]=2

Parametric Formulas  (6)

The volume of an ellipsoid with semimajor axes 3, 2, and 1:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-jwa9dj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-hgiczn
Out[2]=2

The volume of a hemispherical shell in spherical coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-j9ke5e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-zbc7h9
Out[2]=2

The volume of a torus of major radius 5 and minor radius 2:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-1pitot
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-qoz8ox
Out[2]=2

The volume of the product of a disk and a circle embedded in four-dimensional space:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-ur3sdt
Out[1]=1

The volume of the paraboloid over the rectangle :

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-zqfzd3
Out[1]=1

Volume of one octant of a three-sphere using stereographic coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-oienkx
Out[1]=1

Options  (3)Common values & functionality for each option

Assumptions  (1)

The area of an elliptic pyramid with arbitrary semimajor axis , semiminor axis , and height :

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-h12qz7
Out[1]=1

Adding an assumption that the semiaxes are positive simplifies the answer:

In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-yr9vto
Out[2]=2

The region for , , and :

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-24ytqq
Out[3]=3

WorkingPrecision  (2)

Compute the Volume using machine arithmetic:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-777jyw
Out[1]=1

In some cases, the exact answer cannot be computed:

In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-66h1xx
Out[2]=2

Find the Volume using 30 digits of precision:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-5n8cz9
Out[1]=1

Applications  (6)Sample problems that can be solved with this function

A function region :

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-lp42t4
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-4lana5

The region is a volume:

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-t6rz5
Out[3]=3

The volume of the region:

In[4]:=4
✖
https://wolfram.com/xid/0tz9x816-gacd2u
Out[4]=4

Equivalently:

In[5]:=5
✖
https://wolfram.com/xid/0tz9x816-8pyrza
Out[5]=5

Compute the volume of a polyhedron:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-4nvwy6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-l1mcma
Out[2]=2

The shape of the Earth is nearly that of an oblate spheroid with volume:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-70bbro
Out[1]=1

Substitute in the values for the semimajor and semiminor axes:

In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-8q8g1d
Out[2]=2

Find the mass of methanol in a Ball:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-u48x65
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-6bgb0m
Out[2]=2

Find the mean density of a Cone with a non-uniform mass density defined by :

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-vpjoxa
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-qpucm2
Out[2]=2

Compute the volume of empty space in a can with tennis balls, each with a radius of 1.75 inches:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-ug6x97
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-z5qrg4
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-h92vkh
Out[3]=3

Visualize a can of three balls:

In[4]:=4
✖
https://wolfram.com/xid/0tz9x816-ix6xv5
Out[4]=4

Properties & Relations  (5)Properties of the function, and connections to other functions

Volume is a non-negative quantity:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-0v59hv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-76fnat
Out[2]=2

Volume[r] is the same as RegionMeasure[r] for 3D regions:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-nct1tz
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-jblaq
Out[2]=2

Volume[r] is the same as RegionMeasure[r,3] in general:

In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-olwei5
Out[3]=3

Volume[x,s,t,u,c] is equivalent to RegionMeasure[x,{s,t,u},c]:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-oasr4j
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-mvk8pn
Out[2]=2

For a 3D region, Volume is defined as the integral of 1 over that region:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-g4cfg
Out[1]=1

To get the surface volume of a 4D region, use RegionBoundary:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-fnsw8n
Out[1]=1

Possible Issues  (2)Common pitfalls and unexpected behavior

The parametric form of Volume computes the volume of possibly multiple coverings:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-4y4jpk
Out[1]=1

The region version computes the volume of the image:

In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-grcil2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tz9x816-e3v23y
Out[3]=3

The volume of a region of dimension other than 3 is Undefined:

In[1]:=1
✖
https://wolfram.com/xid/0tz9x816-yrdpjb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tz9x816-mz7pc2
Out[2]=2
Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).
✖
Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).

Text

Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).

✖
Wolfram Research (2014), Volume, Wolfram Language function, https://reference.wolfram.com/language/ref/Volume.html (updated 2019).

CMS

Wolfram Language. 2014. "Volume." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Volume.html.

✖
Wolfram Language. 2014. "Volume." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Volume.html.

APA

Wolfram Language. (2014). Volume. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Volume.html

✖
Wolfram Language. (2014). Volume. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Volume.html

BibTeX

@misc{reference.wolfram_2025_volume, author="Wolfram Research", title="{Volume}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Volume.html}", note=[Accessed: 12-May-2025 ]}

✖
@misc{reference.wolfram_2025_volume, author="Wolfram Research", title="{Volume}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Volume.html}", note=[Accessed: 12-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_volume, organization={Wolfram Research}, title={Volume}, year={2019}, url={https://reference.wolfram.com/language/ref/Volume.html}, note=[Accessed: 12-May-2025 ]}

✖
@online{reference.wolfram_2025_volume, organization={Wolfram Research}, title={Volume}, year={2019}, url={https://reference.wolfram.com/language/ref/Volume.html}, note=[Accessed: 12-May-2025 ]}