The ratio of each exterior angle to each interior angle of a regular polygon is 2:3 find the number of sides

Question

Sum of interior angle of a polygon= (2-n)×180

sum of exterior angles of polygons= 360

Answer 1

The measure of each interior angle is (n-2)(180/n) and the measure of each exterior angle is 360/n.

 

So, 2/3 = 360/n ÷ (n-2)(180/n)

 

       2/3 = 360/n · n/(180n-360)

 

       2/3 = 360/[180(n-2)]

 

       2/3 = 2/(n-2)

 

Cross multiply:  2(n-2) = 6

 

                           2n-4 = 6

 

                           2n = 10     So, n = 5

 

The polygon is a regular pentagon.  

Answer 2

If the ratio of each exterior angle to each interior angle is 2:3, then then ratio of the sum of the exterior angles to the sum of the interior angles is also 2:3

 

Be careful, the sum of the interior angles of a polygon is actually (n-2)×180, but I'm going to use (180n-360):

 

3/2=(180n-360)/360

3/2=½n-1

 

5/2=½n

5=n

 

We should double check that a pentagon IS the correct polygon

 

An exterior angle on a regular pentagon is 360/5=72

An interior angle on a regular pentagon is 180-72=108

 

72/108=2/3

72:108=2:3