The problem is that we have n wires of different lengths, i.e. $w_1,w_2,...,w_n$. The wires are aligned in a way such that they enclose the maximum area. What is that maximum area, or its best approximation?
I understand that this would become a cyclic polygon since all the points would lie on a circle for this condition to hold. Is there a mathematical expression that can approximate the area of such a polygon in terms of the lengths only.
I did the following to get the area, but I don't know how to move it further to get a better approximate value in terms of the lengths:
$$Area_{polygon} = Area_{circle} - \sum Area_{circularSegments}$$ $$= \pi r^2 - 1/2r^2(\sum \theta_i - \sum \sin(\theta_i))$$ $$= \pi r^2 - 1/2r^2(2\pi - \sum \sin(\theta_i))$$ $$= r^2/2(\sum \sin(\theta_i))$$
I'd appreciate any help!
