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$ABCD$ is a square with $AB = 8$cm. Arcs $BC$ and $CD$ are semicircles. Express the area of the shaded region, in square centimeters, and in terms of $\pi$. 

 

codehtml127  Jun 29, 2018
 #1
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Where the two semi-circles intersect  in the middle of the square, call this point  E

Connect EC

 

Let the midpoint of  DC  be F

So FC  and FE are radii of the top semi-circle    = 4 cm

And FE  is perpendicular to FC

 

So  CFE is a right triangle  ....and the area of this triangle  = (1/2)FE * FC   =

(1/2) (4) (4)  = 8  cm ^2

 

And  FC , FE  and  arc EC will form  sector FEC  of this top semicircle

And the area of this sector  = (1/2) FE^2 ( pi/2)  = (1/2)  4^2  (pi/2)  = 4  pi cm^2

 

So...the area of the sector  minus  the area of the triangle will  equal 1/2 of the shaded area  =

 

[ 4pi  - 8]  cm^2

 

So...the shaded area  =  2 [ 4pi - 8 ]  = [ 8pi  - 16 ]  cm^2 

 

 

cool cool cool

CPhill  Jun 29, 2018

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