Question

Two times the interior angle of a regular polygon is equal to seven times its exterior angle. Find the interior angle of the polygon and the number of sides in it.

• A. ${130}^{\circ }$ and $n=11$
• B. ${140}^{\circ }$ and $n=9$
• C. ${130}^{\circ }$ and $n=7$
• D. ${140}^{\circ }$ and $n=15$

Answer 1

The correct option is B ${140}^{\circ }$ and $n=9$

Let each exterior angle of a $n$-sided regular polygon be ${}^{\prime }{e}^{\prime }$.

We know $e=\frac{{360}^o}{n}$

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

Given Two times the interior angle of a regular polygon is equal to seven times its exterior angle.

So $2\times$ interior angle of a regular polygon $=7\times e$

$\Rightarrow 2\times \frac{(n-2){180}^o}{n}=7\times \frac{{360}^o}{n}$

$\Rightarrow (n-2){180}^o\times 2=7\times {360}^o$

$\Rightarrow n-2=7$

$\Rightarrow n=7+2$

$\Rightarrow n=9$

So each Angle of a $9$-sided Regular Polygon$=\frac{(9-2){180}^o}{9}=7\times {20}^o={140}^o$