10.401 Find the radius of the circle if the area of the circle is $Q$ square units greater than the area of the inscribed regular dodecagon.
Solution. A regular 12-sided polygon inscribed in a given circle of radius $R$ consists of 12 identical isosceles triangles with legs $R$ and an angle of $30^{\circ}$ at the vertex. Therefore, $Q=\pi R^{2}-\frac{12 R^{2} \sin 30^{\circ}}{2} \Rightarrow R=\sqrt{\frac{Q}{\pi-3}}$.

Answer: $\sqrt{\frac{Q}{\pi-3}}$.