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Algebra Difficulty 4.9 AIME Find the answer

Let P1,P2,…,P6P_{1}, P_{2}, \ldots, P_{6} be points in the complex plane, which are also roots of the equation x6+6x3−216=0.x^{6}+6 x^{3}-216=0. Given that P1P2P3P4P5P6P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} is a convex hexagon, determine the area of this hexagon.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Factor x6+6x3−216=(x3−12)(x3+18).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 123\sqrt[3]{12} and 183.\sqrt[3]{18}. This means our hexagon is composed of 6 triangles, each with sides of length 123\sqrt[3]{12} and 183\sqrt[3]{18} and with a 60 degree angle in between them. This yields the area of each triangle as 332,\frac{3 \sqrt{3}}{2}, so the total area of the hexagon is 9 3.\sqrt{3}.

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