Question

If the difference between an interior angle of a regular polygon of $(n+1)$ sides and an interior angle of a regular polygon of $n$ sides is $4^{\circ }$; find the value of $n$. Also, state the difference between their exterior angles.

• A. $n=9$ and difference between exterior angles $4^{\circ }$
• B. $n=5$ and difference between exterior angles ${22}^{\circ }$
• C. $n=11$ and difference between exterior angles ${12}^{\circ }$
• D. None of these
  

Answer 1

The correct option is A $n=9$ and difference between exterior angles $4^{\circ }$

An interior angle of (n + 1) sided regular polygon = $\frac{{180}^o\left(\right(n+1)-2)}{(n+1)}$

An interior angle of n sided regular polygon = $\frac{{180}^o(n-2)}{n}$

Their difference is $4^o$

So, $\frac{{180}^o\left(\right(n+1)-2)}{(n+1)}-\frac{{180}^o(n-2)}{n}=4^o$

$=>45[\frac{(n-1)}{(n+1)}-\frac{(n-2)}{n}]=1$

$=>45\frac{2}{n(n+1)}=1$

$=>n^2+n-90=0$

$=>(n-9)(n+10)=0$

$=>n=9,-10$

Since n should be a positive number. So, $n=9$