Question

If the centre of a regular hexagon is at origin and one of the vertices on Argand plane is at $1+2i$ and its perimeter is $k\text{units}$, then the numerical value of $k^2$ is

Answer 1

$\angle A_{2}O{A}_3={60}^{\circ },O{A}_2=O{A}_3$, so $△O{A}_2{A}_3$ is equilateral
Now,
$A_2{A}_3=O{A}_3\Rightarrow A_2{A}_3=|1+2i|=\sqrt{5}$

Perimeter
$k=6A_2{A}_3=6\sqrt{5}\Rightarrow k^2=180$