Example 6.3. A square is inscribed in a circle. What is the probability that a point randomly placed in the circle will be inside the square?
Solution. The area of the circle $S=\pi r^{2}$, the area of the square $S_{0}=2 r^{2}$, where $r$ is the radius of the circle (Fig. 69). From here, using formula (2), we find the required probability:

$$
p=\frac{S_{0}}{S}=\frac{2 r^{2}}{\pi r^{2}}=\frac{2}{\pi}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_e915d089cc29fd681be5g-288.jpg?height=352&width=346&top_left_y=610&top_left_x=553)

Fig. 69