Question

The difference between an interior angle of $(n-1)$ sided regular polygon and an interior angle of $(n+1)$ sided regular polygon is $9^{\circ }$. Find the value of $n$.

• A. $3$
• B. $4$
• C. $9$
• D. $12$

Answer 1

Answer: (C)

$9$

Each Angle of a $n$-sided Regular Polygon$=\frac{(n-2){180}^o}{n}$

Each Angle of a $(n-1)$ -sided Regular Polygon$=\frac{\{(n-1)-2\}{180}^o}{n-1}$

Each Angle of a $(n+1)$ -sided Regular Polygon$=\frac{\{(n+1)-2\}{180}^o}{n+1}$

Given that

$\frac{[(n+1)-2]180}{n+1}-\frac{[(n-1)-2]180}{n-1}=9\Rightarrow (\frac{n+1-2}{n+1}-\frac{n-1-2}{n-1})180=9\Rightarrow [(n-1)(n-1)-(n-3)(n+1)]\times 20=(n-1)(n+1)\Rightarrow [(n^2-2n+1)-(n^2+n-3n-3)]\times 20=n^2-1\Rightarrow (n^2-2n+1-n^2-n+3n+3)\times 20=n^2-1\Rightarrow (3+1)\times 20=n^2-1\Rightarrow n^2=80+1\Rightarrow n^2=81\Rightarrow n=9$