A concave polygon has at least one interior angle measuring more than 180°.
Those geometric figures 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 being analyzed.
Concave polygons , in this sense, are figures of this type that have one or more interior angles that measure more than pi 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, one of the segments between two non-consecutive points is outside the polygon, and that is why it is called an exterior diagonal , something that characterizes concave polygons. As expected, this feature complicates certain calculations, such as its surface area, especially in the field of interactive computer applications such as video games.
Analysis of a concave polygon
At first glance, the concave polygon may seem like an extremely complex figure to analyze. However, after inspecting them a little, we notice that they can be decomposed into two or more convex geometric figures , and then the calculations begin to become simpler.
If we can divide a concave polygon into three triangles, for example, it is possible to calculate the surface area of each one by applying one of the following methods, depending on needs:
* The area of any triangle can be obtained by multiplying its base (any of its segments, 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-angled triangles (those that have a 90° angle between two of their sides), the way to understand it in this case is by multiplying its legs (each of the sides that form the right angle mentioned above) with each other and dividing by 2;
* Equilateral triangles (which have sides of equal length to each other) present a slightly greater challenge, since their surface area is calculated by multiplying their height squared by the square root of 3 , over 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, simply add them to obtain the surface of the polygon.
A star figure can be a concave polygon.
Other features and properties
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°.
Example of a concave polygon
The most frequent example of concave polygons are star polygons , which are shaped like a star . 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 met 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 do not measure more than 180° or pi radians and with diagonals that are always interior.