Chapter 4.9, Problem 9P

To determine

To calculate: the formula for the shaded region.

Answer to Problem 9P

  $\pi w\left(2r+w\right)$

Explanation of Solution

Given:

  

Radius of the inner circle = $r$

Width of the outer circle = $w$

Area of circle = $A=\pi r^2$

Calculation:

Radius of the outer circle = $\left(r+w\right)$

Area of outer circle = $\pi {\left(r+w\right)}^2$

Area of inner circle = $\pi r^2$

Formula for shaded region = area of outer circle − area of inner circle

= $\begin{array}{l}\pi {\left(r+w\right)}^2-\pi r^2\\ \pi \left(r^2+2rw+w^2\right)-\pi r^2\\ \pi r^2+2\pi rw+\pi w^2-\pi r^2\\ 2\pi rw+\pi w^2\\ \pi w\left(2r+w\right)\end{array}$

Formula for shaded region is $\pi w\left(2r+w\right)$