Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas.

[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]: 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.

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] 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]: 8 Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate the volume of any object.

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 On 7 April 1795, the metric system was formally defined in French law using six units.

[10] 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.

[12] 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.

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.

The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches.

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.

[5]: 399 Air displacement pipette is used in biology and biochemistry to measure volume of fluids at the microscopic scale.

[15] 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.

[18]: 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] Various other imperial or U.S. customary units of volume are also in use, including:[5]: 396–398 Capacity is the maximum amount of material that a container can hold, measured in volume or weight.

Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns).