10.149. A circle with radius $R$ is divided into two segments by a chord equal to the side of the inscribed square. Determine the area of the smaller of these segments.
## Solution.

Let the side of the inscribed square be denoted by $a$. The diameter of the circle is its diagonal. This means that $2 a^{2}=(2 R)^{2}$, from which $a=R \sqrt{2}$. The area of the circle $S_{1}=\pi R^{2}$. The area of the square $S_{2}=a^{2}=2 R^{2}$. The area of the smaller segment

$$
\begin{aligned}
& S=\frac{1}{4}\left(S_{1}-S_{2}\right)=\frac{\pi R^{2}-2 R^{2}}{4}=\frac{R^{2}(\pi-2)}{4} \\
& \text { Answer: } \quad \frac{R^{2}(\pi-2)}{4}
\end{aligned}
$$