The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as mL or cm³ (milliliters or cubic centimeters).

Volumes of some simple shapes, such as regular, straight-edged and circular shapes can be easily calculated using arithmetic formulas . More complicated shapes can be calculated by integral calculus if a formula exists for its boundary. The volume of any shape can be determined by displacement.

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.

Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure.

Related terms

The density of an object is defined as mass per unit volume. The inverse of density is specific volume which is defined as volume divided by mass.

Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in liters or its derived units), and volume being how much space an object displaces (commonly measured in cubic meters or its derived units).

Volume and capacity are also distinguished in a capacity management setting, where capacity is defined as volume over a specified time period.

In the UK, a tablespoon can also be five fluidrams (about 17.76 mL).

{3} \pi r^3 |r = radius of sphere
which is the integral of the Surface Area of a sphere |- |An ellipsoid: |$\backslash frac\{4\}\{3\}\; \backslash pi\; abc$ |a, b, c = semi-axes of ellipsoid |- |A pyramid: |$\backslash frac\{1\}\{3\}Ah$ |A = area of the base, h = height of pyramid |- |A cone (circular-based pyramid): |$\backslash frac\{1\}\{3\}\; \backslash pi\; r^2\; h$ |r = radius of circle at base, h = distance from base to tip |- |Any figure (calculus required) |$\backslash int\; A(h)\; \backslash ,dh$ |h = any dimension of the figure, A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. This will work for any figure if its cross-sectional area can be determined from h (no matter if the prism is slanted or the cross-sections change shape). |} The units of volume depend on the units of length. If the lengths are in meters, the volume will be in cubic meters.

For their volume formulas, see the articles on tetrahedron and parallelepiped.

Sphere

The volume of a sphere is the integral of infinitesimal circular slabs of thickness $dx$. The calculation for the volume of a sphere with center 0 and radius $r$ is as follows.