The centers of the two circles in the figure are $5 \mathrm{~cm}$ apart, and their radii are $3 \mathrm{~cm}$ and $4 \mathrm{~cm}$. What is the difference in the area of the two shaded regions?

![](https://cdn.mathpix.com/cropped/2024_05_02_341e34ef91138cca6439g-1.jpg?height=347&width=504&top_left_y=385&top_left_x=800)
Let the common area of the two circles be $T$, and the areas of the complementary parts of the circles be $T_{1}<T_{2}$.

The difference in the areas we are looking for is:

$$
T_{2}-T_{1}=\left(T_{2}+T\right)-\left(T_{1}+T\right)=4^{2} \pi-3^{2} \pi=7 \pi \approx 21.99 \mathrm{~cm}^{2}
$$

The result is independent of the distance between the centers of the circles, which of course must be less than the sum of the radii of the circles if we want the two circles to have a common part.