Volume

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Volume
Simple Measuring Cup.jpg
A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
Common symbols
V
SI unit cubic metre
Other units
Litre, fluid ounce, gallon, quart, pint, tsp, fluid dram, in3, yd3, barrel
In SI base units m 3
Extensive?yes
Intensive?no
Conserved?yes for solids and liquids, no for gases, and plasma [lower-alpha 1]
Behaviour under
coord transformation
conserved
Dimension L3

Volume is a measure of regions in three-dimensional space. [1] It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length (cubed) is interrelated with volume. 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. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). [2] [3]

Contents

In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal volume is the hypervolume.

History

Ancient history

6 volumetric measures from the mens ponderia in Pompeii, an ancient municipal institution for the control of weights and measures Pompeji 6 Hohlmasse aus Glas.jpg
6 volumetric measures from the mens ponderia in Pompeii, an ancient municipal institution for the control of weights and measures

The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). [4] :8 The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids, cylinders, frustum and cones. These math problems have been written in the Moscow Mathematical Papyrus (c. 1820 BCE). [5] :403 In the Reisner Papyrus, ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. [4] :116 The Egyptians use their units of length (the cubit, palm, digit) to devise their units of volume, such as the volume cubit [4] :117 or deny [5] :396 (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). [4] :117

The last three books of Euclid's Elements, written in around 300 BCE, detailed the exact formulas for calculating the volume of parallelepipeds, cones, pyramids, cylinders, and spheres. The formula were determined by prior mathematicians by using a primitive form of integration, by breaking the shapes into smaller and simpler pieces. [5] :403 A century later, Archimedes (c.287 – 212 BCE) devised approximate volume formula of several shapes used the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the Middle East and India. [5] :404

Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. [5] :404 Though highly popularized, Archimedes probably does not submerge the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. [6] Instead, he likely have devised a primitive form of a hydrostatic balance. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle. [7]

Calculus and standardization of units

Diagram showing how to measure volume using a graduated cylinder with fluid dram markings, 1926 "How to Measure" diagram, with graduated cylinder measuring fluid drams, 1926.jpg
Diagram showing how to measure volume using a graduated cylinder with fluid dram markings, 1926

In the Middle Ages, many units for measuring volume were made, such as the sester, amber, coomb, and seam. The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. [4] :73–74 In 1618, the London Pharmacopoeia (medicine compound catalog) adopted the Roman gallon [8] or congius [9] as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. [8] Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz). [4] :8

Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate the volume of any object. He devised Cavalieri's principle, which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat, John Wallis, Isaac Barrow, James Gregory, Isaac Newton, Gottfried Wilhelm Leibniz and Maria Gaetana Agnesi in the 17th and 18th centuries to form the modern integral calculus, which remains in use in the 21st century. [5] :404

Metrication and redefinitions

On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the stère  (1 m3) for volume of firewood; the litre  (1 dm3) for volumes of liquid; and the gramme , for mass—defined as the mass of one cubic centimetre of water at maximum density, at about 4 °C (39 °F).[ citation needed ] Thirty years later in 1824, the imperial gallon was defined to be the volume occupied by ten pounds of water at 17 °C (62 °F). [5] :394 This definition was further refined until the United Kingdom's Weights and Measures Act 1985, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. [10]

The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded the metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. [11] The definition of the metre was redefined again in 1983 to use the speed of light and second (which is derived from the caesium standard) and reworded for clarity in 2019. [12]

Properties

As a measure of the Euclidean three-dimensional space, volume cannot be physically measured as a negative value, similar to length and area. Like all continuous monotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to Cavalieri's principle and to the infinitesimal calculus of three-dimensional bodies. [13] A 'unit' of infinitesimally small volume in integral calculus is the volume element; this formulation is useful when working with different coordinate systems, spaces and manifolds.

Volume in general is a Jordan content, so the volume would satisfy these axioms:[ citation needed ]

Measurement

The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches. However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as gourds, sheep or pig stomachs, and bladders. Later on, as metallurgy and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. [5] :393 This method is common for measuring small volume of fluids or granular materials, by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients. [5] :399

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

Some SI units of volume to scale and approximate corresponding mass of water Visualisation litre gram.svg
Some SI units of volume to scale and approximate corresponding mass of water

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

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

Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x). Integral apl rot objem3.svg
Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the volume of rotated f(x).

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:

where and are the plane curve boundaries. [18] :1,3 The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: [18] :6

The volume of a region D in three-dimensional space is given by the triple or volume integral of the constant function over the region. It is usually written as: [19] :Section 14.4

In cylindrical coordinates, the volume integral is

In spherical coordinates (using the convention for angles with as the azimuth and measured from the polar axis; see more on conventions), the volume integral is

Geometric modeling

Low poly triangle mesh of a dolphin Dolphin triangle mesh.png
Low poly triangle mesh of a dolphin

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

See also

Notes

  1. At constant temperature and pressure, ignoring other states of matter for brevity

Related Research Articles

<span class="mw-page-title-main">Area</span> Size of a two-dimensional surface

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

Density is a substance's mass per unit of volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume:

<span class="mw-page-title-main">Integral</span> Operation in mathematical calculus

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

<span class="mw-page-title-main">Litre</span> Unit of volume

The litre or liter is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). A cubic decimetre occupies a volume of 10 cm × 10 cm × 10 cm and is thus equal to one-thousandth of a cubic metre.

Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre.

<span class="mw-page-title-main">Physical quantity</span> Measurable property of a material or system

A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is the algebraic multiplication of a numerical value and a unit of measurement. For example, the physical quantity mass, symbol m, can be quantified as m=n kg, where n is the numerical value and kg is the unit symbol.

<span class="mw-page-title-main">Pressure</span> Force distributed over an area

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

<span class="mw-page-title-main">Relative density</span> Ratio of two densities

Relative density, sometimes called specific gravity, is a dimensionless quantity defined as the ratio of the density of a substance to the density of a given reference material. Specific gravity for liquids is nearly always measured with respect to water at its densest ; for gases, the reference is air at room temperature. The term "relative density" is often preferred in scientific usage, whereas the term "specific gravity" is deprecated.

In chemistry and related fields, the molar volume, symbol Vm, or of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. It is equal to the molar mass (M) divided by the mass density (ρ):

<span class="mw-page-title-main">Cubic centimetre</span> Unit of volume

A cubic centimetre is a commonly used unit of volume that corresponds to the volume of a cube that measures 1 cm × 1 cm × 1 cm. One cubic centimetre corresponds to a volume of one millilitre. The mass of one cubic centimetre of water at 3.98 °C is almost equal to one gram.

<span class="mw-page-title-main">Riemann sum</span> Approximation technique in integral calculus

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule. It can also be applied for approximating the length of curves and other approximations.

<span class="mw-page-title-main">Centroid</span> Mean ("average") position of all the points in a shape

In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the point defined by the arithmetic mean position of all the points in the surface of the figure. In a polytope, it can be found using the arithmetic mean position of the vertices. The same definition extends to any object in -dimensional Euclidean space.

The Method of Mechanical Theorems, also referred to as The Method, is one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures (centroid) and the law of the lever, which were demonstrated by Archimedes in On the Equilibrium of Planes.

<span class="mw-page-title-main">Barrel (unit)</span> Series of units for volume measurement

A barrel is one of several units of volume applied in various contexts; there are dry barrels, fluid barrels, oil barrels, and so forth. For historical reasons the volumes of some barrel units are roughly double the volumes of others; volumes in common use range approximately from 100 to 200 litres. In many connections the term drum is used almost interchangeably with barrel.

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

The number density is an intensive quantity used to describe the degree of concentration of countable objects in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensional linear number density. Population density is an example of areal number density. The term number concentration is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.

<span class="mw-page-title-main">Dilution (equation)</span> Chemistry concept

Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. To dilute a solution means to add more solvent without the addition of more solute. The resulting solution is thoroughly mixed so as to ensure that all parts of the solution are identical.

Dimensional weight, also known as volumetric weight, is a pricing technique for commercial freight transport, which uses an estimated weight that is calculated from the length, width and height of a package.

<span class="mw-page-title-main">Cubic metre</span> SI derived unit of volume

The cubic metre or cubic meter is the unit of volume in the International System of Units (SI). Its symbol is m3. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère, still sometimes used for dry measure. Another alternative name, no longer widely used, was the kilolitre.

<span class="mw-page-title-main">Cavalieri's quadrature formula</span> Mathematical term in calculus

In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral

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