# Thread: Annulus Area Question

1. ## Annulus Area Question

The given chord length 2X is tangent to the inner circle What is the area of the annulus?"

How can I go about understanding this? There's an image of a circle but not sure how to appropriately solve it. I'd appreciate any feedback. Thanks!

2. Hello, LydiaK67!

The given chord length $2x$ 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

$AB$ is a chord of circle $O$ with radius $R.$
It is tangent to the inner circle of radius $r$ at $C.$
. . $AC \,=\,CB\,=\,x$

The area of the annulus is: . $A \;=\;\pi R^2 - \pi r^2 \;=\;\pi(R^2-r^2)$ .[1]

In right triangle $OCB\!:\;\;r^2 + x^2 \:=\:R^2 \quad\Rightarrow\quad R^2 - r^2 \:=\:x^2$ .[2]

Substitute [2] into [1]: . $\boxed{A \;=\;\pi x^2}$