Volume

The volume of a solid body is the amount of ``space'' it occupies. Volume has units of Length cubed (i.e., cm³, m³, in³, etc.) For example, the volume of a box (Rectangular Parallelepiped) of Length $L$, Width $W$, and Height $H$ is given by

$$V=L\times W\times H.$$

The volume can also be computed for irregularly-shaped and curved solids such as the Cylinder and Cube. 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 $r$ denotes the Radius, $h$ the height, $A$ 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).

Surface	Volume
Cone	${\textstyle{1\over 3}}\pi r^2h$
Conical Frustum	${\textstyle{1\over 3}}\pi h({R_1}^2+{R_2}^2+R_1R_2\,)$
Cube	$a^3$
Cylinder	$\pi r^2h$
Ellipsoid	${\textstyle{4\over 3}}\pi abc$
Oblate Spheroid	${\textstyle{4\over 3}}\pi a^2b$
Prolate Spheroid	${\textstyle{4\over 3}}\pi ab^2$
Pyramid	${\textstyle{1\over 3}}Ah$
Pyramidal Frustum	${\textstyle{1\over 3}}h(A_1+A_2+\sqrt{A_1A_2}\,)$
Sphere	${\textstyle{4\over 3}}\pi r^3$
Spherical Cap	${\textstyle{1\over 3}}\pi h^2(3r-h)$
Spherical Sector	${\textstyle{2\over 3}}\pi r^2h$
Spherical Segment	${\textstyle{1\over 6}}\pi h(3a^2+3b^2+h^2)$
Torus	$2\pi^2 Rr^2$

Even simple Surfaces can display surprisingly counterintuitive properties. For instance, the Surface of Revolution of $y=1/x$ around the $x$-axis for $x\geq 1$ is called Gabriel's Horn, and has finite volume, but infinite Surface Area.

The generalization of volume to $n$ Dimensions for $n\geq 4$ is known as Content.

See also Arc Length, Area, Content, Height, Length (Size), Surface Area, Surface of Revolution, Volume Element, Width (Size)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127-132, 1987.