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 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).

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).

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.