In a polygon if *all* the sides are congruent, then it is called is called as a *regular polygon*. Otherwise (even if one side is different) the polygon is a *irregular polygon*.

The following diagram shows the pictures of some common polygons regular and irregular.

## Regular and Irregular Polygons – Common Features

In any polygon whether it is regular or irregular, the number of sides, the number of vertices and the number of angles are all same. Also the number of diagonals of a polygon is given by [n(n – 3)/2], where n is the number of sides.

Another common feature is the sum of the interior angles. Let us draw a irregular polygon and study both the interior and exterior angles of all the vertices.here is a graph from geometry homework help .

Imagine a polygon of *n *number of sides, a part of which is shown above. Suppose you trace around the polygon starting from A to B, B to C and so on and finally come back to point A.

Let x_{k} and y_{k} are the interior and exterior angle at each vertex in general.

If you are tracing the polygon, at every vertex, you have to turn by an angle equal to the exterior angle there. Hence when you reach back the point A, the sum of the angles you have turned is,

y = y_{1} + y_{2} + y_{3} + ………+ y_{n}

But in a curved path you reach the same point means that you have covered 360^{0}.

So, y = y_{1} + y_{2} + y_{3} + ………+ y_{n} = 360^{0}

*In other words, the sum of the exterior angles in a polygon is 360 ^{0}, irrespective of the number of sides.*

Let *x *the sum of all the internal angles of any polygon.

That is, x = x_{1} + x_{2} + x_{3} + ………+ x_{n}

But at any vertex the interior angle and exterior angle are supplementary.

Therefore, x = (180^{0} – y_{1}) + (180^{0} – y_{2}) + (180^{0} – y_{3}) + …….+ (180^{0} – y_{n})

= (n)(180^{0}) – (y_{1} + y_{2} + y_{3} + ………+ y_{n})

= (n)(180^{0}) – (360^{0})

= (180^{0})(n – 2)

**Thus in any polygon of n number of sides, the sum of all interior angles is, (180^{0})(n – 2)**

Some students rewrite (180^{0})(n – 2), as (90^{0})(2n – 4) for easy remembrance as ‘(2n – 4) right angles’.

*In a regular polygon if you divide (180 ^{0})(n – 2) by ‘n’, you get the value of each internal angle. *

## Regular and Irregular Polygons – Measurement of Area

To measure the area of a irregular polygon the only method is to divide the polygon into different known shapes and sum the areas of all the shapes. For example, look at the following irregular quadrilateral.

The following is one of the possible methods to find the area of the quadrilateral ABCD.

By drawing a line segment EC parallel to the base AB, the quadrilateral ABCD can be divided into a trapezoid ABCE and a triangle DEC.

The area of quadrilateral ABCD = Area of the trapezoid ABCE + Area of the triangle DEC