The volume of a solid body is the amount of "space" it occupies. Volume has units of length cubed (i.e., 
, 
,
, etc.) For example, the volume of
a box (cuboid) of length 
, width 
, and height 
is given by
 |
The volume can also be computed for irregularly-shaped and curved solids such as the cylinder and cone. The volume of a surface of revolution is particularly simple to compute due to its symmetry.
The volume of a region can be computed in the Wolfram Language using Volume[reg].
The following table gives volumes for some common surfaces. Here 
denotes the radius,
the height, and 
the base area, and, in the case of
the torus, 
the distance from the torus center to the center of the tube
(Beyer 1987).
| surface | volume |
| cone |  |
| conical frustum |  |
| cube |  |
| cylinder |  |
| ellipsoid |  |
| oblate spheroid |  |
| prolate spheroid |  |
| pyramid |  |
| pyramidal frustum |  |
| sphere |  |
| spherical cap |  |
| spherical sector |  |
| spherical segment |  |
| torus |  |
Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution
of 
around the x-axis
for 
is called Gabriel's
horn, and has finite volume, but infinite surface
area.
The generalization of volume to 
dimensions for 
is known as content.
For many symmetrical solids, the interesting relationship
 |
holds between the surface area 
, volume 
,
and inradius 
. This relationship can be generalized for an arbitrary convex
polytope by defining the harmonic parameter 
in place of the inradius 
(Fjelstad and Ginchev 2003).
Whose Area is the Derivative of the Volume." College
Math. J. 34, 350-358, 2003.Fjelstad, P. and Ginchev, I. "Volume,
Surface Area, and the Harmonic Mean." Math. Mag. 76, 126-129,
2003.