4. Find the area of the figure, the coordinates $(x ; y)$ of the points of which satisfy the system of inequalities

$$
\left\{\begin{array}{l}
|x|+|y| \geq 1 \\
(|x|-1)^{2}+(|y|-1)^{2} \leq 1
\end{array}\right.
$$
Solution. The region defined by the first inequality of the system is the entire plane without the square with vertices at points $(1 ; 0),(0 ; 1),(-1 ; 0),(0 ;-1)$, and the second inequality defines a circle centered at the origin with a radius of 1. Therefore, the desired area is the difference between the area of the circle and the area of the square.

Answer: $\pi-2$.