10.111. The side of a regular triangle inscribed in a circle is equal to $a$. Calculate the area of the square inscribed in the same circle.
## Solution.

Let the radius of the circumscribed circle be denoted by $R$. Then $a=R \sqrt{3}$, from which $R=a / \sqrt{3}$. Since the side of the inscribed square is $R \sqrt{2}$, its area $S=2 R^{2}=2 a^{2} / 3$.

Answer: $2 a^{2} / 3$.