Hello, LydiaK67!
The given chord length $\displaystyle 2x$ is tangent to the inner circle.
What is the area of the annulus? Code:
* * *
* *
* x C x *
A o - - - - * o * - - - - o B
* | * *
* * | * *
* r| *R *
* | * *
* | * *
* - - * - - - - o - - - - * - - *
O
$\displaystyle AB$ is a chord of circle $\displaystyle O$ with radius $\displaystyle R.$
It is tangent to the inner circle of radius $\displaystyle r$ at $\displaystyle C.$
. . $\displaystyle AC \,=\,CB\,=\,x$
The area of the annulus is: .$\displaystyle A \;=\;\pi R^2 - \pi r^2 \;=\;\pi(R^2-r^2)$ .[1]
In right triangle $\displaystyle OCB\!:\;\;r^2 + x^2 \:=\:R^2 \quad\Rightarrow\quad R^2 - r^2 \:=\:x^2$ .[2]
Substitute [2] into [1]: .$\displaystyle \boxed{A \;=\;\pi x^2}$