MathDB
friendly reminder that equilateral hexagons are not regular

Source: 2021 Fall AMC 12A #14

November 11, 2021
AMCAMC 12AMC 12 A

Problem Statement

In the figure, equilateral hexagon ABCDEFABCDEF has three nonadjacent acute interior angles that each measure 30∘30^\circ. The enclosed area of the hexagon is 636\sqrt{3}. What is the perimeter of the hexagon? [asy] size(6cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label("CC",C,2*S); label("DD",D,2*S); label("EE",E,2*S); label("FF",F,2*dir(0)); label("AA",A,2*N); label("BB",B,2*W); [/asy] (<spanclass=′latex−bold′>A</span>) 4(<spanclass=′latex−bold′>B</span>) 43(<spanclass=′latex−bold′>C</span>) 12(<spanclass=′latex−bold′>D</span>) 18(<spanclass=′latex−bold′>E</span>) 123(<span class='latex-bold'>A</span>)\: 4\qquad(<span class='latex-bold'>B</span>) \: 4\sqrt3\qquad(<span class='latex-bold'>C</span>) \: 12\qquad(<span class='latex-bold'>D</span>) \: 18\qquad(<span class='latex-bold'>E</span>) \: 12\sqrt3
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