In everyday life, we often encounter such a characteristic as area. For example - the area of the table, walls, apartment, plot, country, continent. It only applies to flat and conditionally flat surfaces that can be defined by length/width, radius/diameter, diagonals, heights, and angles.

An entire section of geometry is devoted to this, studying plane figures: squares, rectangles, trapezoids, rhombuses, circles, ellipses, triangles - planimetry.

## Historical background

Archaeological studies indicate that the ancient Babylonians were able to measure the surface area 4-5 thousand years ago. It is the Babylonian civilization that is credited with the discovery and implementation of this mathematical characteristic, on which the most complex calculations were subsequently built: from geographical to astronomical.

Initially, area was only used to measure land. They were divided into squares of the same size, which simplified the accounting of croplands and pastures. Subsequently, the characteristic was used in architecture and urban planning.

If in Babylon the concept of "area" was inextricably linked with a square (later - a rectangle), then the ancient Egyptians expanded the Babylonian teaching and applied it to other, more complex figures. So, in ancient Egypt they knew how to determine the area of parallelograms, triangles and trapezoids. Moreover, according to the same basic formulas that are used today.

For example, the area of a rectangle was calculated as its length times its width, and the area of a triangle was calculated as half of its base times its height. When working with more complex figures (polyhedra), they were first broken down into simple figures, and then calculated using basic formulas, substituting the measured values. This method is still used in geometry, despite the presence of special complex formulas for polyhedra.

### Ancient Greece and India

Scientists learned to work with rounded figures only in the III-II centuries BC. We are talking about the ancient Greek researchers Euclid and Archimedes, and in particular about the fundamental work "Beginnings" (books V and XII). In them, Euclid scientifically proved that the areas of circles are related to each other as the squares of their diameters. He also developed a method for constructing a sequence of areas, which, as they grow, gradually "exhaust" the desired area.

In turn, Archimedes for the first time in history calculated the area of a segment of a parabola, and put forward innovative ideas in his scientific work on calculating the turns of spirals. It is to him that the fundamental discovery of inscribed and circumscribed circles belongs, the radii of which can be used to calculate the areas of many geometric shapes with high accuracy.

Indian scientists, having learned from the ancient Egyptians and Greeks, continued their research during the early Middle Ages. So, the famous astronomer and mathematician Brahmagupta in the 7th century AD introduced such a concept as “semiperimeter” (denoted as p), and using it developed new formulas for calculating flat quadrangles inscribed in circles. But all the formulas were presented in the "Metric" and other scientific works not in text, but in graphical form: as diagrams and drawings, and received their final form much later - only in the 17th century, in Europe.

### Europe

Then, in 1604, the exhaustion method discovered by Euclid was generalized by the Italian scientist Luca Valerio. He proved that the difference between the areas of an inscribed and circumscribed figure can be made smaller than any given area, provided they are made up of parallelograms. And the German scientist Johannes Kepler (Johannes Kepler) first calculated the area of the ellipse, which he needed for astronomical research. The essence of the method was to decompose the ellipse into many lines with a step of 1 degree.

As of the 19th-20th centuries, studies of the areas of flat figures were practically exhausted and presented in the form in which they still exist. Only the discovery of Herman Minkowski, who proposed to use an “enveloping layer” for flat figures, which, with a thickness tending to zero, can be considered innovative, makes it possible to determine the desired surface area with high accuracy. But this method only works if additivity is observed, and cannot be considered universal.