9.4. Given a convex polygon $M$ with area $S$ and perimeter $P$. Consider the set $\Phi(x)$ of points that are at a distance of no more than $x$ from $M$ (it is said that a point $A$ is at a distance of no more than $r$ from the polygon $M$ if there exists a point $B$ of the polygon $M$ such that $|A B| \leqslant r$). Find the area of the figure $\Phi(x)$ for each positive number $r$.

AT ThE FactoRy YaRd
9.4. Prove that the geometric locus $\Phi(x)$ consists of the following pieces, as shown in the figure:

a) polygon $M$, b) rectangles of height $x$, adjacent to each side of $M$ from the outside, c) sectors of radius $x$, adjacent to each vertex.

Using the theorem on the sum of the angles of a polygon, prove that the area of the figure $\Phi(x)$ is

$$
s(x)=s+P x+\pi x^{2}
$$

This formula is valid not only for convex polygons but also for any convex figure. Verify it for a circle and

![](https://cdn.mathpix.com/cropped/2024_05_21_20eb9f6ef2c568605a63g-64.jpg?height=453&width=411&top_left_y=512&top_left_x=636)

Fig. 9.4. for a segment.

For non-convex figures, the formula (*) is no longer valid (construct an example), but for them, the area of the figure $\Phi(x)$ is always no greater than $s(x)$.