A circle is inscribed in an equilateral triangle of side  $a$. The area of any square inscribed in this circle is:

Let a be the length of each sides of an equilateral triangle ABC. Then r the radius of the in circle = $\left(\frac{1}{3}\right)$AP (the altitude, median and the angle bisector of angle)
 $\Rightarrow r=\frac{1}{3}\sqrt{a^2-\frac{a^2}{4}}=\frac{a}{2\sqrt{3}}$
Area of the square PQRS inscribed in this circle  
$=P{Q}^2=O{Q}^2=2r^2=2\times \frac{a^2}{4\times 3}=\frac{a^2}{6}$