10.151. A circle is divided into two segments by a chord equal to the side of an inscribed regular triangle. Determine the ratio of the areas of these segments.
Solution.

Let $r$ be the radius of the circle, and $S_{1}$ and $S_{2}$ be the areas of the segments. Then

$$
S_{1}=\frac{\pi r^{2}}{3}-\frac{1}{2} r \cdot \frac{r \sqrt{3}}{2}=\frac{1}{12} r^{2}(4 \pi-3 \sqrt{3}),
$$

$$
\begin{aligned}
& S_{2}=\pi r^{2}-S_{1}=\pi r^{2}-\frac{1}{12} r^{2}(4 \pi-3 \sqrt{3})=\frac{1}{12} r^{2}(8 \pi+3 \sqrt{3}) \\
& \text { Therefore, } \frac{S_{1}}{S_{2}}=\frac{4 \pi-3 \sqrt{3}}{8 \pi+3 \sqrt{3}} \\
& \text { Answer: } \frac{4 \pi-3 \sqrt{3}}{8 \pi+3 \sqrt{3}}
\end{aligned}
$$