Question 1149200
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Consider the diagonals of the square.  Each is a diameter of the circle.  Let r be the radius of the circle.<br>
The diagonals of the square divide the square into four 45-45-90 right triangles.  The side length of the square (the hypotenuse of one of those right triangles) is sqrt(2) times the length of the radius of the circle.<br>
The area of the square is {{{s^2 = (r*sqrt(2))^2 = 2r^2}}}<br>
The area of the circle is {{{(pi)r^2}}}<br>
The ratio of the two areas is<br>
{{{(2r^2)/((pi)r^2) = 2/(pi)}}}<br>