Question 237229
Let r= radius of the circle, and let x = side of the square.


Perimeter of circle = {{{2*pi*r}}}.

Perimeter of rectangle = {{{4x}}}


If the perimeters are equal, then 

{{{2*pi*r=4x}}}


You need to find the ratio of the area of the square to the area of the circle.

Area of circle = {{{pi*r^2}}}
Area of square ={{{x^2}}}


Ratio of area of square to area of circle = {{{(x^2)/(pi*r^2)}}}


Before proceeding, just for fun (and it will be helpful later!!), find the ratio of {{{x/r}}} using the equation above {{{4x=2*pi*r}}}.  Divide both sides by {{{2*pi*r}}}:


{{{ (4x)/(2*pi*r)=1}}}

Divide out the 2 factor:

{{{(2x)/(pi*r)= 1}}}


Divide both sides by 2, and  multiply both sides by {{{pi}}}.

{{{x/r= pi/2}}}.


NOW:


RATIO of AREAS= {{{(Area of Square)/(Area of Circle)}}}
={{{(x^2)/(pi*r^2)=(1/pi)*(x/r)^2}}}
={{{(1/pi)*(pi/2)^2}}}
={{{(1/pi)*(pi^2/2^2)}}}
={{{pi/4}}}


This is quite an interesting problem!  You may want to get a second opinion on it!!  


Dr. Robert J. Rapalje