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Quantity of three dimensional space

Volume is a measure of regions in three dimensional space. 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 and height (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). 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.

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 A century later, Archimedes (c. 287 – 212 BCE) devised approximate volume formula of several shapes using 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. 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.

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. 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.

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

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.