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,

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

Traditional cooking measures

Further information: United States customary units § Units of capacity and volume, and Imperial units § Volume

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).

Volume formulas

Shape	Formula	Variables
A cube	$a^{3}\;$	a = length of any side (or edge)
A rectangular prism:	$l\cdot w\cdot h$	l = length, w = width, h = height
A cylinder:	$\pi r^{2}h\;$	r = radius of circular face, h = height
A general prism:	$B\cdot h$	B = area of the base, h = height
A sphere:	${\frac {4}{3}}\pi r^{3}$	r = radius of sphere which is the integral of the Surface Area of a sphere
An ellipsoid:	${\frac {4}{3}}\pi abc$	a, b, c = semi-axes of ellipsoid
A pyramid:	${\frac {1}{3}}Bh$	B = area of the base, h = height of pyramid
A cone (circular-based pyramid):	${\frac {1}{3}}\pi r^{2}h$	r = radius of circle at base, h = distance from base to tip
Any figure (calculus required)	$\int A(h)\,dh$	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 (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.

Volume formula derivation

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

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

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

See also

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

External links

• Volume calculator - Javascript automatic calculator.