A circle is inscribed in an equilateral triangle of side $6cm$.  Find the area of any square in the circle is

$2x+2y+2z=18$ (Sum of sides)

$\begin{array}{l}x+y+z=9\to \left(1\right)\\ x+z=BC=6\\ \therefore y=3,x=3,z=3\\ x=y=z=3\end{array}$

In $\Delta BOL,\tan 30°=\frac{r}{3}$

$\frac{1}{\sqrt{3}}=\frac{r}{3}\Rightarrow r=\sqrt{3}$

Square $\Rightarrow$ diameter $=$ diagonal of square

$\begin{array}{l}2r=\sqrt{2}a\\ 2\left(\sqrt{3}\right)=\sqrt{2}a\\ \therefore a=\sqrt{6}\end{array}$

$\therefore$ Area of square in the circle $=a^2$

$\begin{array}{l}={\left(\sqrt{6}\right)}^2\\ =6c{m}^2\end{array}$