Someone suggests the following procedure for constructing a square with approximately the same area as a unit radius circle $k$. Let two perpendicular radii of the circle be $O P, O R$, and the intersection of the tangents at $P$ and $R$ be $Q$, and the midpoint of $P Q$ be $S$. The point on $k$ where $O S$ intersects is $T$, and the line parallel to $P Q$ through $T$ intersects $O Q$ at $A$. Then the desired square is the one whose center is $O$ and one of its vertices is $A$.

Determine the error of this procedure (i.e., the difference between the obtained area and the expected area). How much does the approximation improve if $S$ is moved towards $Q$ by $1/44$ of the length of $P Q$?
1. Let $T$ be the projection of $O P-\mathrm{n} U$, so the side of the square in question is $2 \cdot U A=2 \cdot O U$, and its area is $t=4 O U^{2}$. On the other hand, $P S=1 / 2, O S=\sqrt{5} / 2$, and based on the similarity of the right triangles $O T U$ and $O S P$, we have $O U=1 / O S=2 / \sqrt{5}$, thus $t=3.2$ square units.

![](https://cdn.mathpix.com/cropped/2024_05_02_6a2921e409f870fe5cadg-1.jpg?height=626&width=616&top_left_y=305&top_left_x=749)

However, the area $k$ is $11 \pi=3.1415926 \ldots$, so the proposed square's area approximates the circle's area from above with an error of $0.058 \ldots$, which is about $2 \%$ of the area.

2. By shifting the point $S$ to $S'$, we get $P S^{\prime}=23 / 44$, and the area of the new approximating square is calculated similarly to the previous one:

\[
t^{\prime}=4 \cdot O U^{\prime 2}=\frac{4}{O S^{\prime 2}}=\frac{4 \cdot 44^{2}}{44^{2}+23^{2}}=3.1415822 \ldots
\]

This now approximates from below, and its error is less than 0.000011, which is less than $4 \cdot 10^{-6}$ of the area to be approximated. The error of the new approximation is reduced to less than 1/5000 of the previous error.

Maria Boruzs (Tata, Eötvös J. Gymnasium, 2nd year)

[^0]:    ${ }^{1}$ See Kelemenné-Mosonyi-Munkabizottság: Mathematics for the 8th grade of general schools. 4th edition. Tankönyvkiadó, Budapest, 1969. p. 224.