Solution:

The minimum occurs for an equilateral triangle. The area of an equilateral triangle with side-length $s$ is $\frac{\sqrt{3}}{4} s^{2}$, so if the area is $\sqrt{3}$ then $s = \sqrt{\sqrt{3} \frac{4}{\sqrt{3}}} = 2$. Multiplying by $3$ to get the perimeter yields the answer $6$.