# Volume calculator

Volume is one of the most important geometric characteristics: along with the perimeter and area of figures. But it can only be applied to three-dimensional bodies, which are characterized not only by length and width, but also by height/thickness.

Spheres, cubes, cylinders, pyramids, cones, parallelepipeds - all these are three-dimensional figures, the calculation of which is carried out according to special formulas, many of which were discovered by scientists before our era.

## Historical background

### Ancient Egypt and Babylon

The first evidence of the use of three-dimensional figures refers to Ancient Egypt, or rather, to its construction and architecture. Thus, majestic pyramidal structures could not be built without knowing the basic principles for determining mass and volume. This means that the ancient Egyptians, at least, could calculate the volume of cubes, prisms and pyramids.

A vivid example is the tomb of Pharaoh Cheops, 147 meters high, which has an ideal geometric shape of a pyramid. It is impossible to put it together from individual bricks and blocks in such a way that it has stood for more than 4500 years; this requires high-precision mathematical and engineering calculations.

There is no documentary evidence that the ancient Egyptians and Babylonians used specific formulas to calculate volume, and perhaps they were used only in graphic and oral form - following separate principles, not clearly formulated rules.

From Ancient Babylon, only clay tablets have come down to us, which describe the rules for calculating a truncated (not complete) pyramid, but they would not be enough for the construction of objects of such a scale. It is known that many ancient civilizations calculated the volume of elementary figures by multiplying the area of their base by the height, but this is not applicable to such objects as cones, pyramids, tetrahedra. Although they are often found in ancient architecture and have well-defined proportions.

### Ancient Greece

The principles of finding volumes were more clearly formulated in Ancient Greece - from the 5th to the 2nd centuries BC. Euclid introduces the concept of a cube, which simultaneously means both the volume of the figure of the same name and the raising of a number to the 3rd power. And Democritus in the 5th century BC for the first time formulated a rule for finding the volume of a pyramid, which, according to his research, is always equal to 1/3 of the volume of a prism of the same height and with the same base.

In the period from the 6th to the 2nd century BC, ancient Greek mathematicians also learned to calculate the volume of prisms, cylinders and cones, using the already discovered number "pi", which is necessary for calculating all round figures. Archimedes' research formed the basis of the integral method of calculus, and he considered his main discovery to be the formula according to which the volume of a ball is always 2/3 less than the volume of the cylinder described around it. In addition to Archimedes, Democritus and Eudoxus of Cnidus also made a great contribution to the study of geometry.

### New time

During Antiquity, all the basic formulas for calculating three-dimensional figures were derived, and the Middle Ages did not give a single fundamentally new discovery in this area - with the exception of Indian researchers (mainly Brahmagupta), who created several geometric rules in the 6th-7th centuries with the addition of a new value - the semi-perimeter. A fundamentally new approach was applied only in modern times - in the XVI-XVII centuries.

In his work "Geometry" (Geometria indivisibilibus continuorum nova quadam ratione promota) of 1635, the Italian scientist Bonaventura Cavalieri proposed a new principle for finding the volume of a pyramid, and laid the foundation for the further development of mathematics and physics for 300 years to come. The principle is that if at the intersection of two bodies by any plane parallel to some given plane, the cross-sectional areas are equal, the volumes of these bodies are also equal.

It is noteworthy that until the 19th century there were no exact definitions for the volumes of three-dimensional bodies, and they were formulated only in 1887 by Giuseppe Peano, and in 1892 by Marie Enmond Camille Jordan. According to the SI system, the cubic meter became the main unit of measurement of volume, and all other units (ounces, feet, barrels, bushels) remained as alternative ones.

3D geometry aroused particular interest in the 20th century, with the development of abstractionism. In 1966, photographer Charles F. Cochran created his famous “crazy box” photo of an inside-out cube, after which cubic snowflakes, floating, repeating, two-story cubes, and more entered the list of impossible 3D shapes. Modern 3D art is also impossible without the use of generally accepted formulas for finding volume, which, although calculated by a computer, were created many centuries ago.