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Source: 2019 AMC 12B #24

February 14, 2019
2019 AMC 12B2019 AMCAMC 12complex planeareageometry

Problem Statement

Let ω=−12+12i3.\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3. Let SS denote all points in the complex plane of the form a+bω+cω2,a+b\omega+c\omega^2, where 0≤a≤1,0≤b≤1,0\leq a \leq 1,0\leq b\leq 1, and 0≤c≤1.0\leq c\leq 1. What is the area of SS?
<spanclass=′latex−bold′>(A)</span>123<spanclass=′latex−bold′>(B)</span>343<spanclass=′latex−bold′>(C)</span>323<spanclass=′latex−bold′>(D)</span>12π3<spanclass=′latex−bold′>(E)</span>π<span class='latex-bold'>(A) </span> \frac{1}{2}\sqrt3 \qquad<span class='latex-bold'>(B) </span> \frac{3}{4}\sqrt3 \qquad<span class='latex-bold'>(C) </span> \frac{3}{2}\sqrt3\qquad<span class='latex-bold'>(D) </span> \frac{1}{2}\pi\sqrt3 \qquad<span class='latex-bold'>(E) </span> \pi
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