There are an infinite number of flat figures of very different shapes, both right and wrong. A common property of all figures - any of them has an area. The area of โโthe figures is the size of the part of the plane occupied by these figures, expressed in specific units. This value is always expressed by a positive number. The unit of measurement is the area of โโthe square whose side is equal to the unit of length (for example, one meter or one centimeter). The approximate value of the area of โโany figure can be calculated by multiplying the number of unit squares into which it is divided by the area of โโone square.
Other definitions of this concept are as follows:
1. Areas of simple figures are scalar positive quantities that satisfy the conditions:
- for equal figures - equal sizes of areas;
- if the figure is divided into parts (simple figures), then its area is the sum of the areas of these figures;
- a square having a unit of measurement side serves as a unit of area.
2. The squares of figures of complex shape (polygons) are positive values โโthat have the following properties:
- for equal polygons - the same area values;
- if the polygon is composed of several other polygons, its area is equal to the sum of the areas of the latter. This rule holds true for non-overlapping polygons.
As an axiom, it is accepted that the areas of figures (polygons) are positive values.
The definition of the area of โโa circle is given separately as the value to which the area of โโa regular polygon inscribed in the circle of a given circle tends - despite the fact that the number of its sides tends to infinity.
The areas of irregularly shaped figures (arbitrary figures) are not defined, only the methods of their calculation are determined.
The calculation of areas already in antiquity was an important practical task in determining the size of land. The rules for calculating areas several hundred years before our era were formulated by Greek scientists and set forth in Euclidean Principles as theorems. Interestingly, the rules for determining the area of โโsimple shapes in them are the same as at the present time. The areas of geometric figures having a curved contour were calculated using the limit transition.
The calculation of the area of โโsimple figures (triangle, rectangle, square), familiar to everyone from the school bench, is quite simple. It is not necessary to even remember the lettering formulas for the area formula of the figures. Just remember a few simple rules:
1. To calculate the area of โโa square, you need to multiply the length of its side by itself (or elevate it to the second power).
2. The area of โโthe rectangle is calculated by multiplying its length by width. At the same time, it is necessary that the length and width be expressed in the same units.
3. The area of โโa complex figure is calculated by dividing it into several simple ones and adding the resulting areas.
4. The diagonal of the rectangle divides it into two triangles, whose areas are equal and equal to half its area.
5. The area of โโa triangle is calculated as half the product of its height and base.
6. The area of โโa circle is equal to the product of the square of the radius and the well-known number ฯ.
7. The area of โโthe parallelogram is calculated as the product of adjacent sides and the sine of the angle lying between them.
8. The area of โโthe rhombus is ยฝ of the result of multiplying the diagonals by the sine of the inner corner.
9. The area of โโthe trapezoid is found by multiplying its height by the length of the midline, which is equal to the arithmetic mean of the bases. Another option for determining the area of โโa trapezoid is to multiply its diagonals and the sine of the angle lying between them.
For clarity, children in elementary school are often given tasks: to find the area of โโa figure drawn on paper using a palette or a sheet of transparent paper, cut into cells. Such a sheet of paper is superimposed on the measured figure, the number of full cells (units of area) that fit in its contour is considered, then the number of incomplete cells that is divided in half.