12. Overlapping Circles. A circle of radius 15 intersects with a circle of radius 20 at a right angle. Consider the two regions that result after removing their common part from the respective circles. What is the difference in their areas?

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-015.jpg?height=334&width=502&top_left_y=313&top_left_x=777)
12. In general, if we have two intersecting regions with areas equal to $a$ and $b$ respectively, and the area of their

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-079.jpg?height=269&width=488&top_left_y=1696&top_left_x=790)

common part is $x$, then the areas of the corresponding non-overlapping parts are $a-x$ and $b-x$. The difference of these areas is, therefore, $|a-b|$. In our problem, this difference is obviously $\pi(20)^{2}-\pi(15)^{2}=175 \pi$

$[$ [ M. M., 26, 287 (May 1953).]