# Annulus Area Question

• May 19th 2010, 07:24 AM
LydiaK67
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!
• May 19th 2010, 08:05 AM
Soroban
Hello, LydiaK67!

Quote:

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]: .$\displaystyle \boxed{A \;=\;\pi x^2}$