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Need expalnation for this problem: Which is larger: the yellow area or the blue area?

 

 May 8, 2020
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The sum of the yellow regions must be the same as the sum of the blue regions for this reason:

 

The diameter of each of the smaller circles is one-half the diameter of the larger circle.

To make it easy, call the radius of each of the smaller circle "1" so that the radius of the larger circle is "2".

 

The area of the larger circle is:  Area  =  pi · 22  =  4·pi.

 

The area of each of the smaller circles is:  Area  =  pi · 12  =  1·pi..

So, the total area of all the smaller circles (added together) is:  Area  =  4·pi.

 

Therefore, the total area of all the smaller circles is the same as the area of the larger circle.

So, the overlap of the smaller circles (the yellow areas) must equal the "underlap"  -- the area of the larger circle that isn't covered by the smaller circles -- the blue areas.

 May 8, 2020

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