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Volume -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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The volume of a solid body is the amount of "space" it occupies. Volume has units of length cubed (i.e., cm^3, m^3, in^3, etc.) For example, the volume of a box (cuboid) of length L, width W, and height H is given by

V=L×W×H.

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 r denotes the radius, h the height, and A the base area, and, in the case of the torus, R the distance from the torus center to the center of the tube (Beyer 1987).

Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution of y=1/x around the x-axis for x>=1 is called Gabriel's horn, and has finite volume, but infinite surface area.

The generalization of volume to n dimensions for n>=4 is known as content.

For many symmetrical solids, the interesting relationship

S=(dV)/(dr)

holds between the surface area S, volume V, and inradius r. This relationship can be generalized for an arbitrary convex polytope by defining the harmonic parameter h in place of the inradius r (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 R^n 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

Volume

Cite this as:

Weisstein, Eric W. "Volume." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Volume.html

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