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Four quarter circles are drawn inside a unit square, as shown below.  Find the area of the shaded region.

 

 Dec 23, 2020
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See the following :

 

 

Using the Law of Cosines   we  can find EF = one side of a square  which comprises part of the shaded region

 

EF  =  sqrt  [ BE^2  + BF^2  - 2 ( BE * BF)cos (30°)  ]

 

EF  = sqrt [  1^2 + 1^2  - 2 (1 * 1)sqrt (3/2) ] 

 

EF = sqrt  ( 2 - sqrt (3) )

 

EF^2   = area of the square =  (2 -sqrt(3) )

 

And the area  between the circle and the side of the square  =  

 

Area of  30°  sector  of  the circle =  (1/12) pi 1^2  =  pi/12      less

 

Area of triangle BEF  =(1/2) (1)^2 sin (30°)  =  (1/2) * (1/2)  =  1/4

 

So this area  =  pi/12 - 1/4   =   [ pi - 3] / 12

 

And we have 4 of these regions =   [pi -3 ] / 3 =  pi/3 -1

 

So....the total  shaded area  =   (2 -sqrt (3))  + (pi/3 -1)   =

 

[1 + pi/3  - sqrt (3)] units^2  ≈  .315 units^2

 

 

cool cool cool

 Dec 24, 2020

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