Example 1. A square is inscribed in a circle (Fig. 1.2). A point is randomly thrown into the circle. What is the probability that the point will land in the square?
Solution. Let's introduce the following notations: $R$ - the radius of the circle, $a$ - the side of the inscribed square, $A$ - the event of a point falling into the square, $S$ - the area of the circle, $S_{1}$ - the area of the inscribed square. As is known, the area of the circle $S=\pi R^{2}$. The side of the inscribed square in terms of the radius of the circumscribed circle is given by the formula $a=\sqrt{2} R$, so the area of the square $S_{1}=2 R^{2}$.

Assuming in formula (1.5.1) $S_{g}=S_{1}, S_{G}=S$, we find the desired probability

$$
P(A)=\frac{2 R^{2}}{\pi R^{2}}=\frac{2}{\pi} \approx 0.637
$$

Remark. The expression for the side of the square in terms of the radius of the circle can be obtained as follows. From $\triangle K M N$ by the Pythagorean theorem $K N^{2}+N M^{2}=K M^{2}$, i.e.,

$$
\begin{aligned}
& a^{2}+a^{2}=(2 R)^{2}, \quad 2 a^{2}=4 R^{2} \\
& a^{2}=2 R^{2}, a=\sqrt{2} R
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_720b3cbdc89c57e2b629g-027.jpg?height=611&width=597&top_left_y=218&top_left_x=514)

Fig. 1.2