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).
See also
Arc Length, Area, Bellows Conjecture, Cavalieri's Principle, Content, Harmonic Parameter, Height, Length, Surface Area, Surface of Revolution, Volume Element, Volume Theorem, Width Explore this topic in the MathWorld classroom
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References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127-132, 1987.Dorff, M. and Hall, L. "Solids in 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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Volume." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Volume.html