* * *
* *
* 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]: .\(\displaystyle \boxed{A \;=\;\pi x^2}\)