Ratio of the areas of three concentric circles

Question

A target is divided into three regions by concentric circles with radii that are in the ratio $1:2:3$ Find the ratio of the areas of the three regions in the form $a:b:c$ where $a$, $b$ and $c$ are integers

Answer 1

Let $r$ be the smallest radius. Then the radii of the other two circles/disks are $2r$ and $3r$. This means that the areas of these disks are, respectively, $\pi r^{2}$, $\pi (2r)^2=4\pi r^{2}$, and $\pi (3r)^{2}=9\pi r^{2}$. That is, the ratio of the areas is $1$:$4$:$9$.

However, if you are interested in the ratios not of the disks but of the annular rings, then some subtraction is required. The areas of the annular rings are, respectively: $\pi r^{2}$, $4\pi r^{2}-\pi r^{2}=3\pi r^{2}$, and $9\pi r^{2}-4\pi r^{2}=5\pi r^{2}$. Thus, the ratios of the areas of annular regions are $1$:$3$:$5$.