Chapter 4.1, Problem 21MS

Well-rounded shapes. Suppose we have two circles having the same center. The small one has radius r, and the large one has radius R. Let’s now consider two shapes. The first is the doughnut-like region between the two circles. The second is a disk whose diameter is the length of the line segment whose endpoints are on the large circle and whose center point touches the small circle at its north pole. What are the areas of these two shaded regions? How do their sizes compare with each other? (We note that the formula for the area of a circle of radius r equals $\pi {\text{r}}^{\text{2}}.)$