Consider 3 circles, any two of which are externally tangent. Their radii are $r_{1}, r_{2}, r_{3}$. Determine the area enclosed between the 3 circles. Calculate this area numerically if $r_{1}=4, r_{2}=9, r_{3}=36$.
The sought area is obtained by subtracting the sum of the areas of the three circular segments $t_{1}+t_{2}+t_{3}$ that fall within the triangle from the area $T$ of the triangle defined by the centers of the three circles (see Figure 1).

![](https://cdn.mathpix.com/cropped/2024_05_02_eaf48336a4a2edf0765cg-1.jpg?height=596&width=968&top_left_y=266&top_left_x=600)

The sides of the triangle defined by the centers of the circles are $r_{1}+r_{2}, r_{2}+r_{3}, r_{3}+r_{1}$. Using Heron's formula, we get

$$
\begin{gathered}
T=\sqrt{\left(r_{1}+r_{2}+r_{3}\right) r_{1} r_{2} r_{3}} \\
t_{1}+t_{2}+t_{3}=\frac{\pi}{360^{\circ}}\left(r_{1}^{2} \alpha^{\circ}-r_{2}^{2} \beta^{\circ}-r_{3}^{2} \gamma^{\circ}\right)
\end{gathered}
$$

The angles can be calculated unambiguously using the cosine rule. However, if we know, for example, that $r_{1}<r_{2}<r_{3}$, it is simpler to calculate the two smaller (necessarily acute) angles using the area formula:

$$
\sin \beta=\frac{2 T}{\left(r_{2}+r_{1}\right)\left(r_{2}+r_{3}\right)}, \quad \sin \gamma=\frac{2 T}{\left(r_{3}+r_{1}\right)\left(r_{3}+r_{2}\right)}
$$

and the third (possibly obtuse) angle can be determined by simple subtraction.

$$
\alpha=180^{\circ}-(\beta+\gamma)
$$

Substituting the appropriate data:

$T=\sqrt{(4+9+36) 4 \cdot 9 \cdot 36}=\sqrt{49 \cdot 36^{2}}=7 \cdot 36=252$ square units.

$$
\sin \beta=\frac{504}{13 \cdot 45}=\frac{56}{65}, \quad \sin \gamma=\frac{504}{40 \cdot 45}=\frac{7}{25}
$$

Using a four-digit logarithm table,

$$
\beta=59.5^{\circ}, \quad \gamma=16.26^{\circ}, \quad \text { and thus } \quad \alpha=180^{\circ}-75.76^{\circ}=104.24^{\circ},
$$

so

$$
\begin{gathered}
t_{1}+t_{2}+t_{3}=\frac{\pi}{360}(16 \cdot 104.24+81 \cdot 59.5+1296 \cdot 16.26)= \\
=\frac{27560.30 \pi}{360} \sim \frac{2756 \pi}{36}=240.5 \text { square units. }
\end{gathered}
$$

Thus, the sought area $t$ is

$$
t=T-\left(t_{1}+t_{2}+t_{3}\right)=252-240.5=11.5 \text { square units. }
$$

Pál Gavajda (Bp. I., Petőfi g. I. o. t.)