165. Square Field. A square field is fenced with a wooden fence, which is made of 10-meter planks, placed horizontally. The height of the fence is equal to four planks. It is known that the number of planks in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field. [^7]
165. Let's divide the square field into triangles, each of which is bounded by one section (i.e., a segment 10 m long) of the fence and two straight lines connecting the ends of this section with the center of the square. The areas of all such triangles are equal to each other, and their total height is $\frac{1}{2}$ of the side $x$ of our square.

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-129.jpg?height=405&width=416&top_left_y=620&top_left_x=820)

The area of each of them is equal to as many hectares as there are boards in its base, i.e., 4 ha $=40000 \mathrm{~m}^{2}$. Therefore, $\left(\frac{1}{2}\right)\left(\frac{x}{2}\right)(10)=40000$ and, consequently, the side of the square field is $16000 \mathrm{m}=16 \mathrm{km}$.

In general, if the field has the shape of a regular polygon, its area is $\frac{1}{2}$ (perimeter) (radius of the inscribed circle). In our case $4 \cdot \frac{p}{10}=\frac{\frac{1}{2} p r}{10000}$ and $2 r=16000 \mathrm{m}=16 \mathrm{km}$.

[P. P. Poy, Civil Engineering - ASCE, 11, 70 (January 1941).]