Question 217496
perimeter of square equals {{{4*s}}}
perimeter of circle = {{{2*pi*r}}}
-----
we have {{{4*s = 2*pi*r}}} since the perimeters of both the square and the circle are equal.
-----
this makes {{{s = (2*pi*r)/4}}}
-----
area of square equals {{{s^2 = ((2*pi*r)/4)^2 = (4*pi^2*r^2)/16}}}
-----
area of circle equals {{{pi*r^2}}}
-----
ratio of area of square to area of circle equals {{{(4*pi^2*r^2)/(16*pi*r^2)}}}
-----
we simplify this to get:
{{{pi/4}}}
-----
area of square / area of circle = {{{pi/4}}}
-----
this means area of square = area of circle * {{{(pi/4)}}}
-----
this also means area of circle = area of square * {{{(4/pi)}}}
-----
to test, we need a circle with a perimeter the same as a square.
-----
let a side of our square = 7
perimeter of the square is {{{4 * 7 = 28}}}
perimeter of our circle must be 28
this means the radius of our circle equals {{{28/(2*pi) = 4.456338407}}}
this means the area of our circle equals  {{{pi*r^2 = 62.38873769}}}
the area of our square is {{{7^2 = 49}}}
{{{49 * 4/pi = 62.38873769}}}
{{{62.38873769 * pi/4 = 49}}}
-----
the ratio holds.
the answer is that the ratio of the area of the square to the area of the circle is {{{pi/4}}}
-----