SOLUTION: 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, wi

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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.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
Examining this part: (The figure is a circle, with radius r, with another shaded circle surrounding it. The shaded part has a width marked 'w'.)

The shaded region would be surrounding the unshaded smaller circle. You want the area for the shaded region. This will be a difference.

Larger circle area, .
Smaller circle area, .
The shaded region should be the larger region minus the small circle region, leaving just the circular shaded region with the width of w.






-----which should be clear enough.
---commutative property for multiplication.
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

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.


Area of inner (smaller) circle: 

With width of shaded region being “w,” then radius of larger circle = r + w

Area of outer (larger) circle = , or , or 

Area of shaded region = , or  

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