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ABCD is a square with AB = 18 cm. Arcs BC and CD are semicircles. Express the area of the shaded region, in square centimeters, and in terms of pi. 

 

 Dec 4, 2020
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This seems trickier  than it really is

 

Look  at the following figure  :

 

 

Note  that   since the sides of the  square = 18....then DB = sqrt 2*18 = AC

E is the  center of the  square so  BE  = CE  =  sqrt (2)*10

And angle CEB = 90°

And the circle has a radius of 18

 

So....if we take  the area of 1/2 of this  circle  and  subtract  the  area of triangle CEB we will have the same area  as  the shaded region

 

Area of half-circle  with radius of 12  =  pi (6/2)^2  / 2 =  18*pi

Area of triangle CEB  =  area of an isosceles triangle  with equal sides of sqrt (2)/2  and an included angle of 90° =

(1/2) [ sqrt (2) / 2 ] ^2 *sin 90° =  12

 

So....the shaded area =   16*pi - 4

 Dec 5, 2020

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