2021 Fall AMC 12A Problems/Problem 14
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Solution (Law of Cosines and Equilateral Triangle Area)
Isosceles triangles , , and are identical by SAS similarity. by CPCTC, and triangle is equilateral.
Let the side length of the hexagon be . The area of each equilateral triangle is $\frac{1}{2}s^2\sin{30\degree}=\frac{3}{4}s^2$ (Error compiling LaTeX. Unknown error_msg). By the Law of Cosines, the square of the side length of equilateral triangle BDF is . Hence, the area of the triangle is . So the total area of the hexagon is and .