In geometry problems, it is often necessary to calculate the area of a polygon. And it can have a fairly diverse form - from all the familiar triangle to some n-gon with some unimaginable number of vertices. In addition, these polygons are convex or concave. In each specific situation, it depends on the appearance of the figure. So it turns out to choose the best way to solve the problem. The figure may turn out to be correct, which will greatly simplify the solution of the problem.
Some theory about polygons
If we draw three or more intersecting lines, then they form a certain figure. It is a polygon. By the number of intersection points it becomes clear how many vertices he will have. They give the name to the resulting figure. It can be:
Such a figure will certainly be characterized by two provisions:
- The adjacent sides do not belong to the same straight line.
- The non-adjacent have no common points, that is, they do not intersect.
To understand which vertices are adjacent, it will be necessary to see if they belong to one side. If yes, then neighboring. Otherwise, they can be connected by a segment, which must be called a diagonal. They can be carried out only in polygons, which have more than three vertices.
What are their types?
A polygon with more than four angles can be convex or concave. The latter differs in that some of its vertices can lie on opposite sides of a straight line drawn through an arbitrary side of the polygon. In convex always all vertices lie on one side of such a line.
In the school course of geometry, most of the time is devoted to convex figures. Therefore, in problems it is required to know the area of a convex polygon. Then there exists a formula in terms of the radius of the circumscribed circle, which allows finding the desired value for any figure. In other cases, there is no unique solution. For a triangle, the formula is one, and for a square or trapezium completely different. In situations where the figure is incorrect or there are a lot of vertices, it is common to divide them into simple and familiar ones.
What should I do if a figure has three or four vertices?
In the first case, it turns out to be a triangle, and one can use one of the formulas:
A figure with four vertices can be a parallelogram:
The formula for the area of the trapezoid: S = H * (a + B) / 2, where a and B are the lengths of the bases.
What should I do with a regular polygon that has more than four vertices?
To begin with, such a figure is characterized by the fact that in it all sides are equal. Plus, the polygon has the same angles.
If you describe a circle around such a figure, its radius will coincide with a segment from the center of the polygon to one of the vertices. Therefore, in order to calculate the area of a regular polygon with an arbitrary number of vertices, we need the following formula:
From it it is easy to get one that is useful for particular cases:
- triangle: S = (3√3) / 4 * R 2;
- of the square: S = 2 * R 2;
- hexagon: S = (3√3) / 2 * R 2.
The situation with the wrong figure
The way out for finding out the area of a polygon, if it is not right and it can not be attributed to any of the previously known figures, is the algorithm:
- break it into simple figures, for example, triangles so that they do not intersect;
- calculate their area by any formula;
- add up all the results.
What should I do if the coordinates of the vertices of a polygon are given in the problem?
That is, we know a set of pairs of numbers for each point that limit the sides of the figure. Usually they are written as (x1 ; y1 ) for the first, (x2 ; y2 ) - for the second, and the n-th vertex has such values (xn ; yn ). Then the area of the polygon is defined as the sum of n terms. Each of them looks like this: ((yi + 1 + yi ) / 2) * (xi + 1 - xi ). In this expression, i varies from one to n.
It is worth noting that the sign of the result will depend on the bypass of the figure. If you use this formula and move clockwise, the answer will be negative.
Condition. The coordinates of the vertices are given by such values (0.6, 2.1), (1.8, 3.6), (2.2, 2.3), (3.6, 2.4), (3.1, 0.5). It is required to calculate the area of the polygon.
Decision. According to the formula indicated above, the first term will be equal to (1.8 + 0.6) / 2 * (3.6 - 2.1). Here you just need to take the values for the game and X from the second and first points. A simple calculation leads to the result 1.8.
The second term is similarly obtained: (2.2 + 1.8) / 2 * (2.3 - 3.6) = -2.6. When solving such problems, do not be afraid of negative values. Everything goes as it should. This is planned.
Similarly, the values for the third (0.29), the fourth (-6.365) and the fifth summand (2.96) are obtained. Then the total area is: 1.8 + (-2.6) + 0.29 + (-6.365) + 2.96 = - 3.915.
A board for solving a problem for which a polygon is depicted on paper in a cage
Most often puzzling is that in the data there is only the size of the cell. But it turns out that more information is not needed. A recommendation to the solution of this problem is the breaking of a figure into a set of triangles and rectangles. Their area is quite simple to count along the lengths of the sides, which then easily folded.
But often there is a simpler approach. It is to draw the figure to the rectangle and calculate the value of its area. Then count the areas of those elements that were superfluous. Subtract them from the general value. This option sometimes involves a somewhat smaller number of actions.