Two circles have radii of 63 and 73 meters, and the distance between their centers is 63 meters. What is the area of their common part? What fraction of each circle's area is this?
Let the two circles be $k_{1}$ and $k_{2}$, their centers $O_{1}$ and $O_{2}$, their radii $r_{1}=63$ and $r_{2}=73$ meters, and their intersection points $M, N$.

![](https://cdn.mathpix.com/cropped/2024_05_02_eee7ae12ae64b5d615a4g-1.jpg?height=624&width=480&top_left_y=192&top_left_x=801)

Since $O_{1} O_{2}=r_{1}$, $O_{2}$ lies on the circumference of $k_{1}$. The area $t$ of the common region can be obtained by adding the area of the sector $k_{2} O_{2} M N$ to the areas of the smaller segments $O_{2} M$ and $O_{2} N$ of $k_{1}$, or by taking the difference between the area of the sector $O_{1} M N$ and the area of the quadrilateral $\mathrm{O}_{1} \mathrm{MO}_{2} \mathrm{~N}$. Denoting the areas in the same way as the regions themselves:

$$
t=O_{2} M N+O_{1} M N-O_{1} M O_{2} N
$$

In the isosceles triangle $O_{1} O_{2} M$, the bisector of the angle $\omega_{1}$ at $O_{1}$ is perpendicular to the chord $O_{2} M$ of the circle $k_{1}$ with length $r_{2}$, thus

$$
\sin \frac{\omega_{1}}{2}=\frac{73}{2 \cdot 63} \approx 0.5794 \quad \text { so } \quad \omega_{1} \approx 70.80^{\circ}
$$

From this,

$$
O_{1} O_{2} M \varangle=\omega_{2}=90^{\circ}-\frac{\omega_{1}}{2} \approx 54.60^{\circ}
$$

In radians, $\omega_{1} \approx 1.2357, \omega_{2} \approx 0.9529$, and with these, $O_{2} M N=r_{2}^{2} \omega^{2} \approx 5078 \mathrm{~m}^{2}, O_{1} M N=r_{1}^{2} \omega_{1} \approx 4905 \mathrm{~m}^{2}$.

On the other hand, the distance from $O_{1}$ to $O_{2} M$ is $\sqrt{63^{2}-36.5^{2}} \approx 51.35 \mathrm{~m}$, from which $O_{1} M O_{2} N \approx 3749 \mathrm{~m}^{2}$, and finally $t \approx 6234 \mathrm{~m}^{2}$.

The areas of the two circles are: $t_{1} \approx 12470 \mathrm{~m}^{2}, t_{2} \approx 16740 \mathrm{~m}^{2}$, and the common area constitutes 50.0% and 37.2% of these, respectively.

István Filetóth (Nyíregyháza, Vasvári P. g. III. o. t.)

Remark. The ratio $t: t_{1}$, when rounded, indicates that the ratio $r_{2} / r_{1}=73 / 63$ is a good approximation for the so-called "goat tethering problem." This problem is as follows: "Given a circular meadow, a goat can graze half of it if the chain attached to the goat's neck is fixed to a stake at the edge of the meadow. How long can the chain be?"