$A_1A_2A_3….A_{18}$ is a regular 18 sided polygon.B is an external point such that $A_1A_2B$ is an equilateral triangle.If $A_{18}A_1$ and $A_1B$ are adjacent sides of a regular n sided polygon.Then what is $n$.
I found each angle of 18 sided polygon to be $170^\circ$.As $A_{18}A_1$ and $A_1B$ are adjacent sides of a regular n sided polygon.
$\Rightarrow 170^\circ+60^\circ+$each angle of $n$ sided polygon=$360^\circ$
each angle of $n$ sided polygon$=130^\circ$
$\frac{(n-1)\times 180^\circ}{n}=130$ gives $n$=3.6 which is not a natural number.So what should be the value of $n$.Question is correct,no typing mistake.Should answer be equal to 2(which means no $n$ sided polygon. )
Best Answer
The interior angles of a regular $n$-gon are equal to $\,\dfrac{(n-2)\pi}n$, hence the interior angles of an octodecagon are equal to $\,\dfrac{8\pi}9$.
If $BA_1$ and $A_1A_{18}$ are consecutive sides of a regular $n$-gon, its interior angle $\theta$ satisfy the equation: $$\frac\pi3+\frac{8\pi}9+\theta=2\pi,\enspace\text{whence}\quad \theta=\frac{7\pi}9.$$ Thus $$\frac{(n-2)\pi}n=\frac{7\pi}9\iff9n-18=7n\iff n=9.$$