In the picture, you see a square divided into six polygons. All these polygons have vertices at the grid points of a square grid, and the largest of them has an area of $35 \mathrm{~mm}^{2}$. Determine the area of the entire square.

(Bednářová)

![](https://cdn.mathpix.com/cropped/2024_04_17_7f9cac6bd796c2fe4728g-1.jpg?height=417&width=417&top_left_y=1365&top_left_x=842)
First, we number the given polygons. Number 1 will be the polygon in the upper left corner of the square, we number the polygons along the perimeter in a clockwise direction, and number 6 will be the polygon in the center. Their areas expressed in the number of squares are then:

$$
S_{1}=3 ; S_{2}=2.5 ; \quad S_{3}=2.5 ; \quad S_{4}=2.5 ; S_{5}=3.5 ; \quad S_{6}=2
$$

The largest area is that of polygon number 5 (in the lower left corner of the square), which is 3.5 squares. If 3.5 squares have an area of $35 \mathrm{~mm}^{2}$, then 7 squares have an area of $70 \mathrm{~mm}^{2}$. Therefore, 1 square has an area of $70: 7=10 \mathrm{~mm}^{2}$. Since the entire square in the image is composed of 16 such squares, it has an area of $16 \cdot 10=160 \mathrm{~mm}^{2}$.