# The ratio of the number of sides of two regular polygons is 5 : 4 and the difference of their exterior angles is ${9^ \circ }$ . Find the number of sides of both the polygons.

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Hint: Considering n be the greatest common divisor(GCD).Then one polygon has 5n sides,while other has 4n sides and we have to calculate the exterior angle of polygon of 5n-sides and 4n-sides and find their differences

“Complete step-by-step answer:”

Let n be the greatest common divisor (GCD) of the numbers under the question.

Then one polygon has 5n sides, while other has 4n sides

It is well known fact that the sum of exterior angles of each polygon is ${360^ \circ }$

So, the exterior angle of the regular 5n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{5n}}$

Similarly, the exterior angle of the regular 4n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{4n}}$

According to question it is given that difference between the corresponding exterior angles is ${9^ \circ }$

$ \Rightarrow \dfrac{{{{360}^ \circ }}}{{4n}} - \dfrac{{{{360}^ \circ }}}{{5n}} = {9^ \circ }$

$ \Rightarrow \dfrac{{5n - 4n}}{{20{n^2}}} = \dfrac{9}{{360}} = \dfrac{1}{{40}}$

$ \Rightarrow 20n = 40$

$ \Rightarrow n = 2$

So, number of sides in one polygon = $5n = 5 \times 2 = 10$

And number of sides in another polygon $ = 4n = 4 \times 2 = 8$

So this is your answer

NOTE: Whenever we face such a problem the key concept is that we have to remember the exterior angle formula for n sided polygon it will help you in finding your desired answer.

“Complete step-by-step answer:”

Let n be the greatest common divisor (GCD) of the numbers under the question.

Then one polygon has 5n sides, while other has 4n sides

It is well known fact that the sum of exterior angles of each polygon is ${360^ \circ }$

So, the exterior angle of the regular 5n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{5n}}$

Similarly, the exterior angle of the regular 4n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{4n}}$

According to question it is given that difference between the corresponding exterior angles is ${9^ \circ }$

$ \Rightarrow \dfrac{{{{360}^ \circ }}}{{4n}} - \dfrac{{{{360}^ \circ }}}{{5n}} = {9^ \circ }$

$ \Rightarrow \dfrac{{5n - 4n}}{{20{n^2}}} = \dfrac{9}{{360}} = \dfrac{1}{{40}}$

$ \Rightarrow 20n = 40$

$ \Rightarrow n = 2$

So, number of sides in one polygon = $5n = 5 \times 2 = 10$

And number of sides in another polygon $ = 4n = 4 \times 2 = 8$

So this is your answer

NOTE: Whenever we face such a problem the key concept is that we have to remember the exterior angle formula for n sided polygon it will help you in finding your desired answer.

Last updated date: 21st Sep 2023

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