Find the area of the shaded region in square centimeters

Relevant wiki: Area of Triangles - Problem Solving - Easy

Consider my figure.

$\tan 30 = \dfrac{3}{x}$ $\implies$ $x=\dfrac{3}{\tan 30}$

It follows that $AC=2\left(\dfrac{3}{\tan 30}\right)+6=\dfrac{6}{\tan 30}+6$ .

The area of an equilateral triangle is given by $A=\dfrac{\sqrt{3}}{4}a^2$ where $a$ is the side length. The area of a circle is given by $A=\pi r^2$ where $r$ is the radius.

From the figure, the area of the shaded region is equal to the area of the equilateral triangle minus the area of the three circles.

Thus,

$A_{shaded}=\dfrac{\sqrt{3}}{4}\left(\dfrac{6}{\tan 30}+6\right)^2-3(\pi)(3^2) \approx \boxed{32~\text{cm}^2}$ .