If you have two concentric circles as shown below, and the length of the straight line is $4\,\mathrm{cm}$, find the area of the shaded region. Show hint

The area of an annulus is $\pi(R^2 - r^2)$ where $R$ and $r$ are the larger and smaller radii.

Since the straight line is tangent to the smaller circle, it’s perpendicular to a line from the centre. Calling the larger radius $R$ and the smaller radius $r$, this gives us a right-angled triangle with $R$ as the hypotenuse and lengths of $r$ and $4\,\mathrm{cm}\div2$ on the other sides. Using Pythagoras’ theorem, \begin{align*} R^2 &= r^2 + 2^2\\ R^2 - r^2 &= 4 \end{align*} The area of the shaded region is $\pi(R^2 - r^2)$ (see Question 10.8.11) which is $4\pi\,\mathrm{cm^2}$. There’s no need to actually know the radii, which is just as well, because we can’t work them out from the given information.