Let P_{1}, P_{2}, \ldots, P_{6} be points in the complex plane, which are also roots of the equation x^{6}+6 x^{3}-216=0. Given that P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} is a convex hexagon, determine the area of this hexagon.
Solution
Factor x^{6}+6 x^{3}-216=\left(x^{3}-12\right)\left(x^{3}+18\right). This gives us 6 points equally spaced in terms of their angles from the origin, alternating in magnitude between \sqrt[3]{12} and \sqrt[3]{18}. This means our hexagon is composed of 6 triangles, each with sides of length \sqrt[3]{12} and \sqrt[3]{18} and with a 60 degree angle in between them. This yields the area of each triangle as \frac{3 \sqrt{3}}{2}, so the total area of the hexagon is 9 \sqrt{3}.
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