10.153. A circle of radius $R$ is divided by two concentric circles into three figures of equal area. Find the radii of these circles.
## Solution.

Let the radii of the inner circles be denoted by $R_{1}$ and $R_{2}$. Let $S$ be the area of the smallest circle; then $\pi R_{1}^{2}=S, \pi R_{2}^{2}=2 S, \pi R^{2}=3 S$. Therefore, $\pi R_{1}^{2}=\frac{\pi R^{2}}{3}$ and $\pi R_{2}^{2}=\frac{2 \pi R^{2}}{3}$, from which $R_{1}=\frac{R}{\sqrt{3}}$ and $R_{2}=R \sqrt{\frac{2}{3}}$.

Answer: $\frac{R}{\sqrt{3}}$ and $R \sqrt{\frac{2}{3}}$.