Complex Polygon

The point $z = \bigg(\frac{\sqrt{3}}{2} + \frac{i}{2}\bigg)$ is on a regular polygon centered at the origin. This tells us that the polygon is inscribed in a circle of radius $1$ centered at the origin, and $z = \bigg(\frac{\sqrt{3}}{2} + \frac{i}{2}\bigg)$ is a point on the circle in the complex plane. Thus, $x = \frac{\sqrt{3}}{2} = \cos 30^\circ$ and $y = \frac{1}{2}i = i\sin 30^\circ$ . $.$ $\therefore$ The angle between the points $1$ , $0$ and $\bigg(\frac{\sqrt{3}}{2} + \frac{i}{2}\bigg)$ is $30^\circ$ . So each point on the polygon is separated by $30^\circ$ of arc on the unit circle. Thus, the polygon has $\biggl(\frac{360^\circ}{30^\circ}\biggr) = 12$ sides. Therefore, the minimum possible number of sides the polygon has is $12.$ $.$ So A $= 12$ . The area of the dodecadgon ( $12$ sides) is: ${A}_{\circ} = r^2 12\sin\bigg(90^\circ - \frac{180^\circ}{12}\bigg) \bullet \cos\bigg(90^\circ - \frac{180^\circ}{12}\bigg).$ ${A}_{\circ} = 12\sin\big(75^\circ\big)\cos\big(75^\circ\big) = \frac{12}{4} = 3.$ So A $= 12$ and B $= 3$ $.$ $\therefore$ A $+$ B $= \fbox{15}.$