A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. 
The ratio of the area of the triangle to the area of the square is:
$\textbf{(A)}\ \sqrt{3}:1\qquad \textbf{(B)}\ \sqrt{3}:\sqrt{2}\qquad \textbf{(C)}\ 3\sqrt{3}:2\qquad \textbf{(D)}\ 3:\sqrt{2}\qquad \textbf{(E)}\ 3:2\sqrt{2}$
[asy] draw(circle((0,0),100)); draw((173.205,-100)--(-173.205,-100)--(0,200)--(173.205,-100)); draw((-100,0)--(0,100)--(100,0)--(0,-100)--(-100,0)); draw((0,0)--(0,-100),dotted); draw((0,0)--(-173.205,-100),dotted); [/asy]
Let the radius of the circle be $r$.  That means the diameter of the circle is $2r$, so the side length of the square is $r\sqrt{2}$.  Therefore, the area of the square is $2r^2$.
By using 30-60-90 triangles, half of the side length of an [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) is $r\sqrt{3}$, so each side is $2r\sqrt{3}$ units long.  Thus, the area of the equilateral triangle is $3r^2\sqrt{3}$.
The ratio of the area of the equilateral triangle to the area of the square is $\frac{3r^2 \sqrt{3}}{2r^2} = \frac{3 \sqrt{3}}{2}$, so the answer is $\boxed{\textbf{(C)}}$.