Polygon

Mathematics

2022

We explain what a polygon is in geometry, the elements that make it up and what types exist. Also, how your measurements are calculated.

The set of lines of a polygon separates a region of the plane from the rest.

What is a polygon?

In geometry a polygon is called geometric figure plane, composed of a set of line segments connected in such a way as to enclose and delimit a region of the flat, generally without crossing one line with another. Its name comes from the Greek words poly ("a lot and gonos (“angle”), that is to say, that in principle they are geometric figures of numerous angles, although today it is preferred to classify them according to their number of sides and not angles.

polygons are shapes two-dimensional (plane equivalents of three-dimensional polytopes), that is, they have only two dimensions: length and width, and both are determined by the proportions of the lines that compose them. The fundamental thing about a polygon is that the set of its lines separates a region of the plane from the rest, that is, it delimits an "inside" and an "outside", since they are figures closed in on themselves.

There are many types of polygons and many ways of understanding them, depending on whether we are talking about Euclidean or non-Euclidean geometry, but they are usually named depending on the number of sides they have, using numerical prefixes. For example, a pentagon (penta + gonos) is a polygon that has five recognizable sides.

The rest of the polygons are named as follows:

number of sides polygon name
3 trine or triangle
4 tetragon or quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon or octagon
9 nonagon or enneagon
10 Decagon
11 hendecagon or undecagon
12 Dodecagon
13 tridecagon
14 tetradecagon
15 pentadecagon
16 hexadecagon
17 heptadecagon
18 Octodecagon or octadecagon
19 Nonadecagon or enneadecagon
20 isodecagon or icosagon
21 henicosagon
22 Doicosagon
23 Triaicosagon
24 tetraicosagon
25 pentaicosagon
30 Triacontagon
40 tetracontagon
50 Pentacontagon
60 hexacontagon
70 Heptacontagon
80 Octocontagon or Octacontagon
90 Nonacontágono or eneacontágono
100 hectagon
1.000 Chiliagon or kiliagon
10.000 Myriagon

elements of a polygon

Polygons are made up of a series of geometric elements.

The polygons are composed of a series of geometric elements to take into account:

  • sides. They are the line segments that make up the polygon, that is, the lines that trace it on the plane.
  • Vertices. They are the meeting, intersection or union points of the sides of the polygon.
  • Diagonals. They are straight lines that join two non-consecutive vertices within the polygon.
  • Center. Present only in regular polygons, it is a point of its interior area that is equidistant from all its vertices and sides.
  • Interior angles. They are the angles that make up two of its sides or segments in the interior area of ​​the polygon.
  • exterior angles. They are the angles that make up one of its sides or segments in the outer area of ​​the polygon and the projection or continuation of another.

Polygon Types

Polygons are classified in different ways, depending on their specific shape. First of all, it is important to distinguish between regular and irregular polygons:

Regular polygons. They are those whose sides and internal angles have the same measure, being equal to each other. They are symmetrical figures, like the triangle equilateral or square. Furthermore, regular polygons are at the same time:

  • equilateral polygons. They are those polygons whose sides always measure the same.
  • equiangular polygons. They are those polygons whose internal angles always measure the same.

Irregular polygons.They are those whose sides and internal angles are not equal to each other, since they have different measures. For example, a scalene triangle.

On the other hand, polygons can be simple or complex, depending on whether their sides intersect or dry at some point:

  • Simple polygons. They are those whose lines or sides never cross or dry, and therefore have a single outline.
  • complex polygons. They are those that present a crossing or intersection between two or more of their non-consecutive edges or sides.

Lastly, we can distinguish between convex and concave polygons, depending on the general orientation of their shape:

  • convex polygons. They are those simple polygons whose internal angles never exceed 180° of opening. They are characterized in that any side can be contained within the figure.
  • concave polygons. They are those complex polygons whose internal angles exceed 180° of opening. They are characterized in that a straight line is capable of cutting the polygon at more than two different points.

measures of a polygon

Being a flat figure, which exists only in the two-dimensional plane (that is, length and width), but closed in on itself, the polygons contain a segment of the plane and delimit an outside and an inside. Thanks to this, two types of measures:

The perimeter. It is the sum of the length of all the sides of the polygon, and in the case of regular polygons it is calculated by multiplying the length of its sides by the number of these.

The area. It is the portion of the plane delimited by the sides of the polygon, that is, its "interior" area. Its calculation, however, requires different procedures, for example:

  • In a triangle, it is calculated by multiplying the base and height and dividing by 2.
  • In a regular quadrilateral (square), it is calculated by squaring the length of any of its sides.
  • In a right quadrilateral (rectangle), it is calculated by multiplying its base by its height.

Which plane figures are not polygons?

Not all plane figures are polygons. Those figures that do not close on themselves (that is, that do not have an interior area), that have curved lines in their formation or whose non-consecutive sides intersect, should not be considered as polygons.

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