In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. Or simply a closed plane figure bounded by three or more line segment to form a closed loop is called a polygon.p1


  • The line segments forming the polygon are called sides.
  • The point of intersection of two line segments is called a vertex.
  • Number of vertices of a polygon is equal to the number of line segments or sides.

Different types of Polygons:  This is based on the number of sides that the polygon has. Here are few examples:P10

p4Diagonal of a Polygon: A line segment joining any two non-consecutive vertices is called

a diagonal of the polygon.

The dotted lines are diagonals of the shown polygons.

Interior and Exterior Angles of a Polygon: This is an important concept.

\angle 1 \ and\  \angle 2  are interior angles. These are made by the two sides of the polygon.p5

\angle 3  is called an exterior angle. This is formed by extending a side of the polygon as shown in the adjacent figure.

Just by looking at the figure you can tell that

\angle 2  +   \angle 3 = 180^{\circ} 

Hence we can say that:

Exterior Angle + Adjacent Interior Angle = 180^{\circ} 

Convex Polygon vs Concave Polygonp6

If the interior angle of the polygon is less than = 180^{\circ} , then it is called convex polygon. If you look at any of the polygons shown above, you will see that all the interior angles are less than = 180^{\circ} .

But there can be cases where the interior angle of a polygon could be more than = 180^{\circ} . Take a look at the adjacent figure. Here you will see that \angle 1 > 180^{\circ}  (which is a reflex angle).

Regular Polygon: A polygon that satisfies the following condition is called a regular polygon.

  1. All sides are equal
  2. All interior angles are equal
  3. All exterior angles are equal

For a regular polygon with n sides we have the following:

\displaystyle \text{Each Interior Angle }=  \Bigg[\frac{(2n-4)\times 90^{\circ}}{n} \Bigg]^{\circ} 


A polygon can be divided into (n-2) triangles.

See a few examples in the adjacent figure.l52

We know that the sum of the angles of a triangle is = 180^{\circ} .

\text{Therefore the sum of the interior angles }= (n-2) \times 180 = (2n - 4) \times 90^{\circ}

\Rightarrow \displaystyle \text{Interior Angle } =   \frac{(2n-4)\times 90^{\circ}}{n} 

\displaystyle \text{Each Exterior Angle } =   \Big[\frac{360}{n} \Big]^{\circ} 


\text{Sum of all Interior Angles + Sum of all Exterior Angles } = n \times 180 = n \times (2 \times 90)^{\circ}

\text{Sum of all Exterior Angles }= n \times (2 \times 90)^{\circ} -  (2n - 4) \times 90^{\circ}=360^{\circ}

\displaystyle \text{Each Exterior Angle } =   \Big[\frac{360}{n} \Big]^{\circ} 

From the above point 2, we can also say that

\displaystyle n =  \Big[\frac{360}{\text{Each Exterior Angle}} \Big]^{\circ} 

We already know that

\text{Exterior Angle + Adjacent Interior Angle   }= 180^{\circ}

\Rightarrow \text{Exterior Angle } =180^{\circ} - \text{Adjacent Interior Angle}