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Which is larger: the yellow area or the blue area?

 

 Dec 25, 2019
 #1
avatar+106516 
+1

Call the radius  of the small circles, R

Call the radius of the large circle, 2R

 

(1/2)  of one of the  yellow areas = [ (1/4)pi R^2  -  (1/2)R^2  ]

 

So   the total of the yellow area  = 8 [ (1/4)pi R^2   - (1/2) R^2]  = R^2 [ 2pi - 4] 

 

And  one of the pink areas  =   pi R^2  - [ pi R^2  - 2 R^2]   =  2R^2

So....the total of the pink areas =  8R^2

 

And the  area  of the large circle is  pi (2R)^2 =  4 pi R^2

 

So....the  blue area  =

 

Area of large circle -  pink areas  - yellow area

 

   4 pi R^2 - 8R^2  - R^2 [ 2pi - 4]  =

 

4pi R^2  - 8R^2  - 2pi R^2  + 4R^2  =

 

2 pi R^2  - 4R^2

 

R^2  [ 2pi  - 4 ] 

 

So....the areas are equal   !!!

 

 

 

cool cool cool

 Dec 26, 2019
 #2
avatar+141 
+1

Large circle radius   r = 2          Area of a large circle:     A = 22 * pai = 12.566 u2

Small circle radius   r = 1          Area of 4 small circles:   A = pai * 4   =  12.566 u2

This proves that the blue and yellow areas are equal. indecision

 Dec 28, 2019

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