Volume

Ancient history , an ancient municipal institution for the control of weights and measures The precision of volume measurements in the ancient period usually ranges between . The Egyptians use their units of length (the cubit, palm, digit) to devise their units of volume, such as the volume cubit 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. Calculus and standardization of units 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. or congius as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Thirty years later in 1824, the imperial gallon was defined to be the volume occupied by ten pounds of water at . 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. 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. 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. == 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. 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. == 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. 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. Air displacement pipette is used in biology and biochemistry to measure volume of fluids at the microscopic scale. 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 of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in a reservoir, the container's volume is modeled by shapes and calculated using mathematics. 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. Therefore, volume has a unit dimension of L3. 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). 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. == Computation ==

Basic shapes that the volume of a cone is a third of a cylinder of equal diameter and height 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:V = \pi \int_a^b \left| f(x)^2 - g(x)^2\right|\,dxwhere f(x) and g(x) are the plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: \iiint_D 1 \,dx\,dy\,dz. In cylindrical coordinates, the volume integral is \iiint_D r\,dr\,d\theta\,dz, In spherical coordinates (using the convention for angles with \theta as the azimuth and \varphi measured from the polar axis; see more on conventions), the volume integral is \iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi . Geometric modeling 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 ==

• Density is the substance's mass per unit volume, or total mass divided by total volume. • Specific volume is total volume divided by mass, or the inverse of density. • 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 ==