What will be the area of shaded region (grey colored) if the given figure is square of side 6cm?

enter image source here

1 Answer
CW
Jul 30, 2017

shaded area #=6*(3sqrt3-pi) ~~ 12.33 " cm"^2#

Explanation:

enter image source here
See the figure above.

Green area #=#area of sector #DAF# - yellow area

As #CF and DF# are the radius of the quadrants,
#=> CF=DF=BC=CD=6#
#=> DeltaDFC# is equilateral.
#=> angleCDF=60^@#
#=> angleADF=30^@#
#=> EF=6sin60=6*sqrt3/2=3sqrt3#

Yellow area = area of sector #CDF- #area #DeltaCDF#
#= pi*6^2*60/360-1/2*3sqrt3*6#
#= 6pi-9sqrt3#

Green area = #=#area of sector #DAF# - yellow area
#=pi*6^2*30/360-(6pi-9sqrt3)#
#= 3pi-(6pi-9sqrt3)#
#= 9sqrt3-3pi#

Hence, the shaded area #A_s# in your figure #= 2xx# green area
#=> A_s=2*(9sqrt3-3pi)#
#=18sqrt3-6pi=6(3sqrt3-pi)~~12.33 " cm" ^2#

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