Perimeter is a characteristic that can be attributed to any flat (and not only) figure. Its boundaries can be characterized by the perimeter, or the sum of the lengths of the sides. This characteristic is also suitable for many volumetric figures.
Definition and general characteristics
In geometry, the perimeter is denoted by the capital Latin letter "P" - from the Latin word perimeter, which, in turn, comes from the ancient Greek περίμετρον (circle). This characteristic was used even before our era, and allowed to determine the boundaries of land and other flat surfaces.
All shapes that contain angles - starting with a triangle and ending with complex polyhedra - can be represented as lines, which are indicated by capital Latin letters in alphabetical order: a, b, c, d, and so on. Thus, the sum of the sides of a triangle will always be expressed as a + b + c, and trapezoids - as a + b + c + d.
The sides of a flat polygon can also be represented as segments between two points, which are denoted by capital Latin letters: AB, BC, CD, and so on. Regardless of the notation used, the perimeter is always equal to the sum of the lengths of the sides, and is considered in the same units.
Historical background
The need to calculate perimeters arose in ancient times - when it was necessary to delimit land plots. Subsequently, this characteristic was used in architecture and construction: when laying foundations and calculating the required amount of building materials.
It is known that the perimeter of a circle in ancient Egypt was calculated back in the 15th-14th centuries BC. For this, a constant was used, today known as the number "pi" (π) and equal to 3.14 ... Although it received its modern name and designation much later - in 1706.
The ancient Egyptians knew up to 10 decimal places in the number π: 3.1415926535..., while modern science knows 100 trillion digits. Nevertheless, even two signs (3.14) are enough to calculate the circumference with a sufficiently high accuracy. And the length of a circle, in fact, is also its perimeter, respectively: P = 2πr, or P = πd. These formulas, but with different notation, were known to the ancient Egyptians over 3500 years ago.
Much later, in the 6th-5th centuries BC, the ancient Greek scientist Pythagoras indirectly used trigonometry to find perimeters.
Since knowing all the sides of a triangle is a prerequisite for finding the perimeter, unknown sides can be found using known angles. For this, Pythagoras used the sine - the ratio of the opposite leg to the hypotenuse, and the cosine - the ratio of the adjacent leg to the hypotenuse. Having thus calculated the desired length of the side, it can be included in the expression P = a + b + c and find out the perimeter of the triangle.
And in the 3rd-2nd centuries BC, the no less famous ancient Greek scientist Archimedes found a way to determine the perimeters by approximation: using regular polygons described around a circle.
Correlation with area
Studies of the perimeters of geometric figures were carried out in parallel with the calculations of their areas. Despite the common belief that the larger the area, the larger the perimeter, these characteristics are not related in any way. For example, if you take a rectangle with a width of 0.001 arbitrary units and a length of 1000 units, its perimeter will be 2000, and for a rectangle with a width of 0.5 and a length of 2 it will be equal to 5. In this case, the area of \u200b\u200bboth rectangles will be equal to one.
The situation with multi-structure figures looks even clearer. The reverse pattern is observed in them: the larger the perimeter, the smaller the area, and vice versa. In the 5th century AD, this became the reason for the uneven distribution of sown areas among the peasants. Not knowing about this pattern, they divided the plots along the perimeters, and not according to the areas, although the amount of the harvested crop is always proportional to the area, not the perimeter. The ancient philosopher Proclus Diadoch, the head of the Platonic Academy, wrote about this.
A little later, in the 6th century AD, India introduced the definition of the semi-perimeter, a value that is now denoted in formulas by the capital letter “p”. It is used to calculate the areas of many geometric shapes and can greatly simplify their writing. As the name implies, to calculate the semiperimeter, you need to add the lengths of all sides of the figure and divide the result by two.
It is not known for certain who and when for the first time in history began to use such a characteristic as a perimeter for practical purposes. It already existed in ancient Egypt, but it is not a fact that it was the Egyptians who invented and put it into circulation. Throughout the subsequent history of civilizations, it was widely used in geometric formulas, and today it is one of the fundamental characteristics, along with area and volume.
