Question 1104011
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Let s be the side length of the regular octagon that is formed.  Then the 4 corners that are cut off of the original square can be put together to form a square of side length s; so the area of the 4 corners cut off is s^2; so the area of the octagon is x^2-s^2.<br>
To find the relationship between s and x, observe that
{{{x = s/sqrt(2)+s+s/sqrt(2)}}}
{{{x = s(sqrt(2)+1)}}}
{{{s = x/(sqrt(2)+1) = x*(sqrt(2)-1)}}}<br>
Then the area of the octagon is
{{{x^2-s^2 = x^2 - (x*(sqrt(2)-1))^2 = x^2 - x^2(3-2*sqrt(2)) = x^2-3x^2+2x^2*sqrt(2) = x^2(2*sqrt(2)-2) = 2x^2(sqrt(2)-1)}}}