Volume

As a

Air displacement pipette is used in biology and biochemistry to measure volume of fluids at the microscopic scale.^[14] Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories. There, volume of liquids is measured using graduated cylinders, pipettes and volumetric flasks. The largest of such calibrated containers are petroleum storage tanks, some can hold up to 1,000,000 bbl (160,000,000 L) of fluids.^[5]: 399 Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made.^[5]: 403

For even larger volumes such as in a reservoir, the container's volume is modeled by shapes and calculated using mathematics.^[5]: 403 The task of numerically computing the volume of objects is studied in the field of computational geometry in computer science, investigating efficient algorithms to perform this computation, approximately or exactly, for various types of objects. For instance, the convex volume approximation technique shows how to approximate the volume of any convex body using a membership oracle.^{[citation needed]}

Units

To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m³). The cubic metre is also a SI derived unit.^[15] Therefore, volume has a unit dimension of L³.^[16]

The metric units of volume uses metric prefixes, strictly in powers of ten. When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm³ = 2.3 (cm)³ = 2.3 (0.01 m)³ = 0.0000023 m³ (five zeros).^[17]: 143

Commonly used prefixes for cubed length units are the cubic millimetre (mm³), cubic centimetre (cm³), cubic decimetre (dm³), cubic metre (m³) and the cubic kilometre (km³). The conversion between the prefix units are as follows: 1000 mm³ = 1 cm³, 1000 cm³ = 1 dm³, and 1000 dm³ = 1 m³.^[1] The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm³ = 1000 cm³ = 0.001 m³.^[17]: 145 For the litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L.^[1]

Litres are most commonly used for items (such as fluids and solids that can be poured) which are measured by the capacity or size of their container, whereas cubic metres (and derived units) are most commonly used for items measured either by their dimensions or their displacements.^{[citation needed]}

Various other imperial or U.S. customary units of volume are also in use, including:^{[5]: 396–398}

• cubic inch, cubic foot, cubic yard, acre-foot, cubic mile;
• minim, drachm, fluid ounce, pint;
• teaspoon, tablespoon;
• gill, quart, gallon, barrel
  ;
• cord, peck, bushel, hogshead.

Capacity and volume

Capacity is the maximum amount of material that a container can hold, measured in volume or weight. However, the contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of fuel oil will not be able to contain the same 7,200 t (15,900,000 lb) of naphtha, due to naphtha's lower density and thus larger volume.^{[5]: 390–391}

Computation

Basic shapes

For many shapes such as the cube, cuboid and cylinder, they have an essentially the same volume calculation formula as one for the prism: the base of the shape multiplied by its height.

Integral calculus

The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution, by rotating a plane curve around a line on the same plane. The washer or disc integration method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as:

$${\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx}$$

${\textstyle f(x)}$

${\textstyle g(x)}$

$${\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx}$$

${\displaystyle f(x,y,z)=1}$ over the region. It is usually written as:

$${\displaystyle \iiint _{D}1\,dx\,dy\,dz.}$$

In cylindrical coordinates, the volume integral is

$${\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,}$$

In spherical coordinates (using the convention for angles with ${\displaystyle \theta }$ as the azimuth and ${\displaystyle \varphi }$ measured from the polar axis; see more on conventions), the volume integral is

$${\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi .}$$

Geometric modeling

A polygon mesh is a representation of the object's surface, using polygons. The volume mesh explicitly define its volume and surface properties.

Derived quantities

• Density is the substance's mass per unit volume, or total mass divided by total volume.^[20]
• Specific volume is total volume divided by mass, or the inverse of density.^[21]
• The volumetric flow rate or discharge is the volume of fluid which passes through a given surface per unit time.
• The volumetric heat capacity is the heat capacity of the substance divided by its volume.

See also

• Baggage allowance
• Banach–Tarski paradox
• Dimensional weight
• Dimensioning

Notes

References