In the figure A1, is the area bounded by a square and a quarter circle with center a vertex of the square. A2 is the area bounded by the square and the circle. What is the ratio of A1 to A2?

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1 Answer
CW
Jun 20, 2018

#(A_1)/(A_2)=(2pi)/(pi-2)#

Explanation:

enter image source here
Let #s# be the side length of the square.
Let #O and r# be the center and the radius of the circle, respectively, as shown in the figure.
#=># as #angleAOB=90^@, => s=sqrt2r#
#A_1=#Yellow area #=(pis^2)/4=(pi(sqrt2r)^2)/4=1/2pir^2#
#A_2=#Green area #=# area of sector #OAB -# area of #DeltaOAB#
#=(pir^2)/4-r^2/2#
#=((pi-2)r^2)/4#

#(A_1)/(A_2)=((pir^2)/2)/(((pi-2)r^2)/4)=(2pi)/(pi-2)~~5.504#

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