The answer, using the "Sometimes the heart can see what’s invisible to the eye " Theorem, is
[tex]A = \bf{36}\sqrt{3} - 12 \pi[/tex]
Proof of this theorem:
[tex]A = 2 \left(\frac{1}{2}(6)(6 \tan(60)\right) + \pi(6)^2 \cdot \frac{120}{360}\ \ \blacksquare [/tex]
