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Volume is the quantity of three-dimensional space bounded by a closed surface, for example, the space that occupies or contains a substance (solid, liquid, gas, or plasma). Volume is often measured in cubic meters in SI units. Volume usually refers to the capacity of the container; for example, the amount of fluid (gas or liquid) that a container can hold, as compared to the amount of space that the container itself displaces. Three dimensional mathematical shapes are also given volumes. The volumes of some simple shapes, such as regular, straight, and round, can be easily calculated using arithmetic formulas. The volumes of complex shapes can be calculated using integral calculus if there is a formula for the shape boundary. One-dimensional shapes (as lines) and two-dimensional shapes (as squares) have zero volume in three-dimensional space.

The volume of a solid (regular or irregular) can be determined by displacing the liquid. Liquid displacement can also be used to determine gas volume. The total volume of the two substances is usually greater than the volume of only one of the substances. However, sometimes one substance will dissolve in another, and in such cases, the combined volume is not additive.

In differential geometry, volume is expressed in terms of the volume form and is an important global Riemannian invariant. In thermodynamics, volume is the main parameter and the associated pressure variable. The Oxford English Dictionary defines capacity as “a measure applied to the contents of a vessel and to liquids, grain, etc., taking the form of that in which they are located” (the word “capacity” has other unrelated meanings such as incapacity management.) The capacity is not identical in meaning to volume; although closely related, the capacity of a container is always the volume in its interior. Units of capacity are the SI liter and its derived units and Imperial units such as gill, pint, gallon, and others. In the SI system, the units of volume and capacity are closely related: one liter is equal to 1 cubic decimeter, the capacity of a cube with a side of 10 cm. In other systems, the transformation is nontrivial; the capacity of a vehicle’s fuel tank is rarely reported in cubic feet, for example, but in gallons (an imperial gallon fills a volume of 0.1605 cu ft).

The density of an object is defined as the ratio of mass to volume. Inverse density is the specific volume, defined as volume divided by mass. Specific volume is an important concept in thermodynamics, where the volume of the working fluid is often an important parameter of the system under study.

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