A4. A circle with radius 2 rolls around a regular pentagon with side 4. During this process, the circle remains pressed against the pentagon until it returns to its initial position.

Calculate the area of the region that is covered by the circle during this revolution.

![](https://cdn.mathpix.com/cropped/2024_04_17_6ccf422d4ecbc0dab674g-1.jpg?height=448&width=462&top_left_y=1666&top_left_x=1294)
A4. The longest line segment within a circle is a diameter. During the rolling, there is always a diameter that determines the distance to the point that is farthest from the pentagon at that moment. As long as the circle rolls along a side, the outer boundary is thus a line segment parallel to the side. When the circle rolls around a vertex, the diameter describes a circular arc. The area of the region covered by the circle is therefore equal to five squares with side length 4 and five circular sectors (pie slices) that together form a circle with radius 4. The required

![](https://cdn.mathpix.com/cropped/2024_04_17_774e242ba19fa9593e58g-1.jpg?height=452&width=475&top_left_y=2110&top_left_x=1293)
area is: $5 \times 4^{2}+\pi \times 4^{2}=16(5+\pi)$.