In problems on geometry it is often required to calculate the area of a polygon. Moreover, it can have a rather diverse shape - from the familiar triangle to a certain n-gon with some unimaginable number of vertices. Moreover, these polygons are convex or concave. In each situation it is supposed to build on the appearance of the figure. So it turns out to choose the best way to solve the problem. The shape may be correct, which will significantly simplify the solution of the problem.
A bit of polygon theory
If three or more intersecting straight lines are drawn, then they form a certain figure. That it is a polygon. By the number of intersection points, it becomes clear how many vertices it will have. They give the name of the resulting figure. It may be:
Such a figure will certainly be characterized by two provisions:
- Adjacent parties do not belong to one straight line.
- Non-adjacent points have no common points, that is, they do not intersect.
To understand which vertices are adjacent, you need to see if they belong to the same side. If so, the next. Otherwise, they can be connected by a segment, which must be called a diagonal. They can be drawn only in polygons with more than three vertices.
What are their types?
A polygon with more than four corners may be convex or concave. The difference of the latter is that some of its vertices may lie on opposite sides of a straight line drawn through an arbitrary side of a polygon. In a convex one, all vertices always lie on one side of such a line.
In the school course of geometry, most of the time is given to exactly convex figures. Therefore, in problems it is required to find out the area of a convex polygon. Then there is a formula through the radius of the circumscribed circle, which allows you to find the desired value for any shape. In other cases, a unique solution does not exist. For a triangle, the formula is one, and for a square or trapezoid completely different. In situations where the shape is irregular or there are a lot of peaks, it is customary to divide them into simple and familiar ones.
What to do if a piece has three or four vertices?
In the first case, it will be a triangle, and you can use one of the formulas:
A figure with four vertices can be a parallelogram:
The formula for the trapezoid area is S = n * (a + b) / 2, where a and b are the lengths of the bases.
What to do with a regular polygon with more than four vertices?
For a start, such a figure is characterized by the fact that in it all sides are equal. Plus, the polygon has the same angles.
If a circle is described around such a figure, then its radius will coincide with the 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 special cases:
- the triangles: S = (3√3) / 4 * R 2;
- square: S = 2 * R 2;
- hexagon: S = (3√3) / 2 * R 2.
The situation with the wrong figure
An exit for how to find out the area of a polygon, if it is not correct and cannot be attributed to any of the previously known figures, is the algorithm:
- break it into simple shapes, such as triangles, so that they do not intersect;
- calculate their area using any formula;
- add up all the results.
What to do if the coordinates of the vertices of the polygon are given in the problem?
That is, a set of pairs of numbers for each point is known 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 traversal of the shape. When using the specified formula and moving in a clockwise direction, the answer will be negative.
Condition. The coordinates of the vertices are given by the values (0.6; 2.1), (1.8; 3.6), (2.2; 2.3), (3.6; 2.4), (3.1; 0.5). Required to calculate the area of the polygon.
Decision. According to the formula above, the first term will be (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 will lead to the result of 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), fourth (-6.365) and fifth terms (2.96) are obtained. Then the total area is: 1.8 + (-2.6) + 0.29 + (-6.365) + 2.96 = - 3.915.
Council for solving the problem for which the polygon is depicted on paper in the cell
Most often, it is puzzling that only the cell size is present in the data. But it turns out that more information is not needed. The recommendation to solve this problem is to divide the shape into many triangles and rectangles. Their area is quite simple to count the length of the parties, which are then easy to fold.
But often there is a simpler approach. It consists in drawing a figure to a rectangle and calculating the value of its area. Then count the areas of those elements that were superfluous. Subtract them from the total. This option sometimes involves a slightly smaller number of actions.