PROOF of the formula for SUM OF INTERIOR ANGLES OF A POLYGON

SUM OF INTERIOR ANGLES OF A POLYGON

A polygon is any 2-dimensional shape formed with straight lines. 

Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. 

The name tells you how many sides the shape has. For example, a triangle has three sides, and a quadrilateral has four sides.

We all learnt in our math class that the sum of the interior angles of a polygon can be calculated by using the formula : (n-2) x 180 , where n is the number of sides of the polygon.




But How did this formula come to be ? 
Lets learn it step by step:

• We know that the sum of all the angles of a triangle is equal to 180:

• Now lets draw a square(a polygon)

• Choose a point on one of its sides and draw lines to join with all other sides (only one in case of a square) 

• Hence, two triangles are formed as shown below. Therefore, as one triangle has sum of angles of 180, the sum of angles of the square would be : 2 x 180 = 360

Now, lets draw a five sided polygon (a pentagon):

• Select a point on one of its sides.

• Draw lines joining the other points on the other sides

• Note that three triangles are formed inside the pentagon. Hence the sum of the interior angles would be :  

3 x 180= 540

Lets look at another example: 

• This an 8 sided polygon (an octagon) 

• Starting from a point and drawing lines joining it with the other points, we get 6 triangles in the polygon.

• So, the sum of the angles would be: 

6 x 180 = 1080

Have you noticed a relation between the polygons and the number of triangles that can be formed inside them? 

Basically, the number of triangles that can be formed is 2 less than the number of sides of a polygon! 

We saw that 2 triangles were formed in a 4-sided polygon i.e square, 3 triangles were formed in a 5 sided polygon i.e a pentagon, 6 triangles were formed in an 8 sided polygon i.e an octagon. 

Similarly, n-2  triangles would be formed in an n sided polygon.

Therefore, the sum of the interior angles of a polygon would be:

        Sum of int angles = No. of triangles x 180

        Sum of int angles  = (n-2) x 180 

Hence proved!