3. Let there be two concentric circles, and let a chord of the larger circle that is tangent to the smaller circle have a length of 2 cm.

Calculate the area of the circular ring!
Solution. Let point $O$ denote the center of the given circles, $AB$ the given chord, and $D$ the midpoint of $AB$.

The radii of the circles are $R=\overline{O A}$ for the larger circle and $r=\overline{O D}$ for the smaller circle. By applying the Pythagorean theorem to $\triangle A D O$, we get $R^{2}-r^{2}=\overline{A D}^{2}=1$. The area

![](https://cdn.mathpix.com/cropped/2024_06_05_aa5de80eeceb04c02556g-6.jpg?height=415&width=417&top_left_y=1162&top_left_x=1034)
of the annulus is

$$
P=R^{2} \pi-r^{2} \pi=\left(R^{2}-r^{2}\right) \pi=1 \cdot \pi=\pi \mathrm{cm}^{2}
$$