The difference between an exterior angle of $(n - 1)$ sided regular polygon and an exterior angle of $(n + 2)$ sided regular polygon is $6^{\circ}$ . Find the value of $n$ .

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Solution

$T o t a l s u m o f e x t e r n a l a n g l e s o f n - s i d e d r e g u l a r p o l y g o n = 360^{o} E x t e r n a l a n g l e o f (n - 1) - s i d e d r e g u l a r p o l y g o n = \frac{360}{n - 1} E x t e r n a l a n g l e o f (n + 2) - s i d e d r e g u l a r p o l y g o n = \frac{360}{n + 2} G i v e n t h a t \frac{360}{n - 1} - \frac{360}{n + 2} = 6 \Rightarrow (\frac{1}{n - 1} - \frac{1}{n + 2}) \times 360 = 6 \Rightarrow (\frac{1}{n - 1} - \frac{1}{n + 2}) \times 60 = 1 \Rightarrow \frac{(n + 2) - (n - 1)}{(n - 1) (n + 2)} \times 60 = 1 \Rightarrow (n + 2 - n + 1) \times 60 = (n - 1) (n + 2) \Rightarrow n^{2} + 2 n - n - 2 = 3 \times 60 \Rightarrow n^{2} + n - 182 = 0 B r e a k i n g t h e m i d d l e t e r m i n s u c h a w a y t h a t p r o d u c t w i l l b e (- 182) & d i f f e r e n c e w i l l b e 1 \Rightarrow n^{2} + 14 n - 13 n - 182 = 0 \Rightarrow n (n + 14) - 13 (n + 14) = 0 \Rightarrow (n + 14) (n - 13) = 0 S o n = 13 o r n = - 14 N o w n i s a n a t u r a l n u m b e r s o i t c a n n o t b e n e g a t i v e . T h e r e f o r e n = 13$

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The difference between an exterior angle of (n−1) sided regular polygon and an exterior angle of (n+2) sided regular polygon is 6∘. Find the value of n.

The difference between an exterior angle of $(n - 1)$ sided regular polygon and an exterior angle of $(n + 2)$ sided regular polygon is $6^{\circ}$ . Find the value of $n$ .