10.123. The area of an isosceles trapezoid circumscribed around a circle is $S$. Determine the lateral side of the trapezoid, given that the acute angle at the base is $\pi / 6$.
## Solution.

Let $x$ be the length of the lateral side; then the height of the trapezoid is $\frac{1}{2} x$. Since the trapezoid is circumscribed around a circle, the sum of its bases is equal to the sum of the lateral sides. Therefore, the area of the trapezoid $S=\frac{1}{2} \cdot 2 x \cdot \frac{1}{2} x$, from which $x=\sqrt{2 S}$.

Answer: $\sqrt{2 S}$.