Volume

For other uses, see Volume (disambiguation).

Volume is how much three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains, often quantified YOU MAMMA;) 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.

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 litres 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 capacity management, where capacity is defined as volume over a specified time period.

Volume formulas

Shape	Volume formula	Variables
Any figure (calculus required)	$V=\int A(h)\,dh$	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).
Cube	$a^{3}\;$	a = length of any side (or edge)
Cylinder	$\pi r^{2}h\;$	r = radius of circular face, h = height
Prism	$B\cdot h$	B = area of the base, h = height
Rectangular prism	$l\cdot w\cdot h$	l = length, w = width, h = height
Sphere	${\frac {4}{3}}\pi r^{3}$	r = radius of sphere which is the integral of the Surface Area of a sphere
Ellipsoid	${\frac {4}{3}}\pi abc$	a, b, c = semi-axes of ellipsoid
Pyramid	${\frac {1}{3}}Bh$	B = area of the base, h = height of pyramid
Cone	${\frac {1}{3}}\pi r^{2}h$	r = radius of circle at base, h = distance from base to tip
Tetrahedron	${{\sqrt {2}} \over 12}a^{3}\,$	edge length $a$
Parallelpiped	$V=abc{\sqrt {K}}$ ${\begin{aligned}K=&1+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}$	a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges

The units of volume depend on the units of length. If the lengths are in meters, the volume will be in cubic meters.

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.

The radius of the circular slabs is $r={\sqrt {y^{2}+x^{2}}}$

The surface area of the circular slab is $\pi r^{2}$.

The volume of the sphere can be calculated as $\int _{-r}^{r}\pi (y^{2}+x^{2})\,dx=\int _{-r}^{r}\pi y^{2}\,dy+\int _{-r}^{r}\pi x^{2}\,dx.$

Now $\int _{-r}^{r}\pi y^{2}\,dy=2\pi {\frac {r^{3}}{3}}$ and $\int _{-r}^{r}\pi x^{2}\,dx=2\pi {\frac {r^{3}}{3}}.$

Combining yields $\left({\frac {2}{3}}+{\frac {2}{3}}\right)\pi r^{3}={\frac {4}{3}}\pi r^{3}.$

This formula can be derived more quickly using the formula for the sphere's surface area, which is $4\pi r^{2}$. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to

$\int _{0}^{r}4\pi u^{2}\,du$ = ${\frac {4}{3}}\pi r^{3}.$

Cone

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) 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, $r{\frac {(h-x)}{h}}.$

The surface area of the circular slab is then $\pi \left(r{\frac {(h-x)}{h}}\right)^{2}=\pi r^{2}{\frac {(h-x)^{2}}{h^{2}}}.$

The volume of the cone can then be calculated as $\int _{0}^{h}\pi r^{2}{\frac {(h-x)^{2}}{h^{2}}}dx,$

and after extraction of the constants: ${\frac {\pi r^{2}}{h^{2}}}\int _{0}^{h}(h-x)^{2}dx$

Integrating gives us ${\frac {\pi r^{2}}{h^{2}}}\left({\frac {h^{3}}{3}}\right)={\frac {1}{3}}\pi r^{2}h.$

Traditional volumetric cooking measures

Further information: United States customary units and imperial units

measure	US	Imperial	metric
teaspoon	1/6 U.S. fluid ounce (about 4.929 mL)	1/6 Imperial fluid ounce (about 4.736 mL)	5 mL
tablespoon = 3 teaspoons	½ U.S. fluid ounce (about 14.79 mL)	½ Imperial fluid ounce (about 14.21 mL)	15 mL
cup	8 U.S. fluid ounces or ½ U.S. liquid pint (about 237 mL)	10 Imperial fluid ounces or ½ Imperial pint (about 284 mL)	250 mL

In the UK, a tablespoon can also be five fluidrams (about 17.76 mL). In Australia, a tablespoon is 4 teaspoons (20 mL).

See also

• Orders of magnitude (volume)
• Length
• Perimeter
• Area
• Mass
• Weight
• Conversion of units
• Dimensional weight
• Dimensioning
• Volume form
• Volume (thermodynamics)
• Banach–Tarski paradox

External links

• Volume calculator - Javascript automatic calculator.