A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? $\textbf{(A)}\ 1\qquad$ $\textbf{(B)}\ \frac{\sqrt{6}}{2}\qquad$ $\textbf{(C)}\ \sqrt{3} \qquad$ $\textbf{(D)}\ 2\qquad$ $\textbf{(E)}\ \frac{3\sqrt{2}}{2}$