## 52. Math Puzzle $9 / 69$

A passage from one of Jack London's novels provides us with material for a geometric calculation.

"In the middle of a square field stood a steel mast, deeply embedded in the ground. From the top of the mast, a steel cable ran to a tractor, where it was attached. The tractor driver threw the lever, and the engine began to work.

Now the tractor moved forward, describing a circle around the mast as the center.

Graham commented: 'To make the setup work better, the only thing left is to transform the circle into a square.'

'Right, on a square field, much of the land remains unplowed with our current method.'

Graham did the calculations and then said: 'We lose about three acres on a plot of ten acres.'"

Now, are you in agreement with the solution?
The calculation is incorrect.

The loss is below 0.3 of the ground. Assuming the side of a square is $a$; its area is therefore $a^{2}$. The diameter of the inscribed circle is also $a$ and its area $\frac{\pi}{4} a^{2}$. The unused part of a field square thus amounts to

$$
a^{2}-\frac{\pi}{4} a^{2}=\left(1-\frac{\pi}{4}\right) a^{2} \approx 0.22 a^{2}
$$

From this, it follows that the unworked part of the arable land is not $30 \%$, as Graham assumed, but only $22 \%$.