Those figures in geometry that are flat and are made up of straight, non-aligned segments are called polygons. Within this classification, it is possible to find a large number of varieties that depend on the characteristics that are analyzed.
The concave polygons, in this sense, are the figures of this type having one or more interior angles measuring more than radians or 180 °. These polygons, on the other hand, have one or more diagonals that are exterior.
The diagonal of the polygon is defined as the union of two non-consecutive vertices of the figure. In this case, as can be seen in the second image, one of the segments between two non-consecutive points is outside the polygon, and that is why we speak of an exterior diagonal, something that characterizes concave polygons. As expected, this feature complicates certain calculations, such as its surface, especially in the field of interactive computer applications such as video games.
At first glance, the concave polygon can seem an extremely complex figure to analyze; The same goes for the two shown in the images in this article. However, after a little inspection, we note that they can be decomposed into two or more convex geometric figures, and then the calculations begin to become simpler.
Let's take the polygon in the first image, for example: with little effort, we can divide it into three triangles. Having done this, it is possible to calculate the area of each one by applying one of the following methods, as needed:
* The area of every triangle can be obtained by multiplying its base (any of its segments, which are obtained by joining two of its vertices) by its height (the distance between the midpoint of the base and the remaining vertex) and then dividing the result For 2;
* Although the previous formula also works for right triangles (those that have an angle of 90 ° between two of their sides), the way to understand it in this case is by multiplying their legs (each of the sides that form the right angle above) by each other and dividing by 2;
There are more ways to specify the surface of a triangle, but it is also possible to find squares within a concave polygon, something that makes things even easier, since in this case you simply have to multiply its smaller side by the larger one. Once all the surfaces have been calculated, just add them together to obtain that of the polygon.
Another characteristic of concave polygons is that they always have two or more vertices that, linked by a segment, will intersect at least one of the sides of the figure.
Due to these properties, triangles (which are polygons that have three sides) can never be concave since their interior angles never exceed pi radians or 180 °.
The most common example of concave polygons are star polygons, which are star- shaped. As can be confirmed by analyzing this class of polygons, they have at least one internal angle with more than 180 ° and an exterior diagonal.
When these properties are not fulfilled and the figures cannot be classified within the group of concave polygons, they enter the set of convex polygons.
As opposed to concave polygons, therefore, convex polygons can be defined as those with internal angles that measure no more than 180 ° or pi radians and with diagonals that are always interior.