Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. The volumes of more complicated shapes can be calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.
Units
Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre (cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
- 1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres,
so
- 1 cubic metre = 1000 litres.
Small amounts of liquid are often measured in millilitres, where
- 1 millilitre = 0.001 litres = 1 cubic centimetre.
Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, and the hogshead.
Related terms
Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units).
Volume and capacity are also distinguished in capacity management, where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity.
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. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.
The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second ).
Volume formulas
| Shape | Volume formula | Variables |
|---|---|---|
| Cube | a = length of any side (or edge) | |
| Cylinder | r = radius of circular face, h = height | |
| Prism | B = area of the base, h = height | |
| Rectangular prism | l = length, w = width, h = height | |
| Sphere | r = radius of sphere which is the integral of the surface area of a sphere |
|
| Ellipsoid | a, b, c = semi-axes of ellipsoid | |
| Pyramid | B = area of the base, h = height of pyramid | |
| Cone | r = radius of circle at base, h = distance from base to tip | |
| Tetrahedron | edge length | |
| Parallelepiped | 
|
a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges |
| Any volumetric sweep (calculus required) |
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. a and b are the limits of integration for the volumetric sweep. (This will work for any figure if its cross-sectional area can be determined from h). |
|
| Any rotated figure (washer method) (calculus required) |
and are functions expressing the outer and inner radii of the function, respectively. | |
| Klein bottle | No volume—it has no inside. |
Ratio of volumes of a cone, sphere and cylinder of the same radius and height
The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is
the volume of the sphere is
while the volume of the cylinder is
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.
Volume formula derivations
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.
A tattered copy of Johnson’s large Dictionary was a great delight to me, on account of the specimens of English versifications which I found in the Introduction. I learned them as if they were so many poems. I used to keep this old volume close to my pillow; and I amused myself when I awoke in the morning by reciting its jingling contrasts of iambic and trochaic and dactylic metre, and thinking what a charming occupation it must be to “make up” verses.
— Lucy Larcom (1824–1893)
The surface area of the circular slab is .
The radius of the circular slabs, defined such that the x-axis cuts perpendicularly through them, is;
or
where y or z can be taken to represent the radius of a slab at a particular x value.
Using y as the slab radius, the volume of the sphere can be calculated as
Now
Combining yields gives
Measured by any standard known to science—by horse-power, calories, volts, mass in any shape,—the tension and vibration and volume and so-called progression of society were full a thousand times greater in 1900 than in 1800; Mthe force had doubled ten times over, and the speed, when measured by electrical standards as in telegraphy, approached infinity, and had annihilated both space and time. No law of material movement applied to it.
— Henry Brooks Adams (1838–1918)
This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to
=
Cone
The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies cones as well. But for an explanation using calculus:
The volume of a cone is the integral of infinitesimal circular slabs of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.
The radius of each circular slab is r if x = 0 and 0 if x = h, and varying linearly in between—that is,
The surface area of the circular slab is then
The volume of the cone can then be calculated as
and after extraction of the constants:
Integrating gives us