Lesson Types of Polygon and properties

In this lesson we are going to discuss the concept of Types of Polygon and its properties.

We will start with the basic understanding of a polygon and will look into the all basic types and properties and issues related to them.

A Polygon is a closed plane figure made up of several line segments that are joined together. The sides do not cross each other and exactly two sides meet at every vertex.

The word polygon is a combination of two Greek words: "poly" means many and "gon" means angle.

A polygon is named by the number of sides it has, example "Tri" means "three," so the simplest polygon is called the triangle , because it has three angles. It also has three sides and three vertices. This way all higher order polygons can be named.

Different types of Polygon according to geometry are :

1.Regular Polygon- In this polygon all angles are equal and all sides are of the same length. Regular polygons are both equiangular and equilateral.

2.Irregular Polygon- Here each side may a different length, each angle may be a different measure. This is a opposite of a regular poygon.

3.Equiangular Polygon- In this polygon all angles are equal. If all sides are also equal then its a regular polygon.

4.Equilateral Polygon- In this polygon all sides are of the same length. If all angles are also equal then its a regular polygon.

5.Convex Polygon- A straight line drawn through a convex polygon crosses at most two sides. In it every interior angle is less than 180° and all vertices 'point outwards' away from the interior. The Regular polygons are always convex.

6.Concave Polygon- In it at least one straight line can be drawn through a concave polygon that crosses more than two sides. In it at least one interior angle is more than 180° and some vertices push 'inwards' towards the interior of the polygon in a concave polygon.

7.Crossed Polygon- A polygon where one or more sides crossed back over itself is called a crossed polygon. Most of the properties and theorems concerning polygons do not apply to this shape.

The basic properties of all Polygons (regular and irregular) are:

1.Interior angles: The interior angles of a polygon are those angles at each vertex on the inside of the polygon. There is one per vertex. So for a polygon with N sides, there are N vertices and N interior angles.

For a regular polygon all the interior angles are the same.For an irregular polygon , each angle may be different.

The interior angle is calculated by the following general formula,

Sum of all Interior Angles
The sum of all interior angles is independent of its geometry.The formula of calculating it is,



where, n is number of sides in a polygon.

For a regular polygon of n sides, the total angle is is spread evenly among all the interior angles, since they all have the same values.Hence the formula for calculating interior angle of a regular polygon is given by,



2.Exterior Angles : The angle formed on the outside of a polygon between a side and the extended adjacent side.

3.Diagonals : The diagonals of a polygon are lines linking any two non-adjacent vertices.

A diagonal of a polygon is a line segment joining two vertices. While creating a general formula for calculating number of diagonals we will have to keep in mind following isues,

From any given vertex, there is no diagonal to the vertex on either side of it, since that would lay on top of a side. Also, there is obviously no diagonal from a vertex back to itself. This means there are three less diagonals than there are vertices. (diagonals to itself and one either side are not counted).

As described above, the number of diagonals from a single vertex is three less the the number of vertices or sides, or (n-3).

As there are N vertices,hence total number of diagonals are

But each diagonal has two ends, so this would count each one twice. So as a final step we divide by 2, for the final formula:



Properties of a Regular Polygon

1.Inradius: The Inradius/apothem of a polygon is a line from the center to the midpoint of a side. This is also the inradius - the radius of the incircle.

2.Circumradius: The radius of a regular polygon is a line from the center to any vertex. It is also the radius of the circumcircle of the polygon.

3.Incircle: The incircle is the largest circle that will fit inside a regular polygon. Its radius is the inradius of the polygon.

4.Circumcircle: The circle that passes through all the vertices of a regular polygon. Its radius is the radius of the polygon.

For further reading on polygons refer to Wikipedia.