Question 876802

Hi,

The area of a circle of radius r is given by the formula A=πr^2. Use this fact to find the area for the shaded area in the figure below. 

(The figure is a circle, with radius r, with another shaded circle surrounding it. The shaded part has a width marked 'w'.) 

The answer given is A=2πrw + πw^2. Can you please explain how this is done.

Thanks.
<pre>
Area of inner (smaller) circle: {{{pi*r^2}}}
With width of shaded region being “w,” then radius of larger circle = r + w
Area of outer (larger) circle = {{{pi(r + w)^2}}}, or {{{pi(r^2 + 2rw + w^2)}}}, or {{{pi*r^2 + 2pi*rw + pi*w^2}}}
Area of shaded region = {{{pi*r^2 + 2pi*rw + pi*w^2 - pi*r^2}}}, or {{{highlight_green(highlight_green(2pi*rw + pi*w^2))}}}