Question

The angles of a polygon are in $A.P.$ The smallest angle is $\frac{2{\pi }^c}{3}$ and common difference is $5^o$, then the number of sides are:

• A. $7$
• B. $9$
• C. $8$
• D. $11$

Answer 1

The correct option is B $9$

Sum of all $n$ interior angles of polygon $=(n-2){180}^o$

smallest angle $={120}^o$ common difference $5o$

sum $=\frac{n}{2}[2120+(n-1\left)\right]5$

$\frac{n}{2}[2\times 120+(n-1)\times 5]=(n-2)\times 180$

$\frac{n}{2}[5n+235]=(n-2)\times 180$

$5n^2+235n=360n-720$

$5n^2-125m+720=0\Rightarrow n^2-25+144=0$

$(n-16)(n-9)=0$

$n=16,9$

$n=16$ then ${16}^{th}$ angle $=120+15\times 5=195$

Hence $=n=9$