algebraically is's not that difficult.
what you have is a circle with a radius of 3 centimeters.
since the area of a circle is equal to pi * r^2, then the area of this circle is 9 * pi.
you are told that the area of the circle minus the area of the square is equal to 7 * pi.
you are told that the length of a side of the square is x.
that means the area of the square is x^2.
therefore, the area of the circle minus the area of the square must be equal to 9 * pi - x^2.
if the net area is 7 * pi, then you get:
9 * pi - x^2 = 7 * pi
subtract 7 * pi from both sides of this equation and you get:
9 * pi - 7 * pi - x^2 = 0
this reduces to 2 * pi - x^2 = 0
this is the formula you are looking for.
if you solve this formula for x^2, you will x^2 = 2 * pi.
the area of the square inside the circle is 2 * pi.
that makes each side of the square equal to sqrt(2 * pi).
the perimeter of the square is 4 * the length of each side of the square.
that makes the perimeter of the square equal to 4 * sqrt(2 * pi).
you can play games with this answer, but it will be the same.
for example:
4 = sqrt(16), therefore the answer becomes sqrt(16) * sqrt(2 * pi) which then becomes sqrt(16 * 2 * pi) which becomes sqrt(32 * pi).
if i did this right, 4 * sqrt(2 *pi) should be equal to sqrt(32 * pi).
i used my calculator to confirm that they are, indeed, equal to each other.
so, your answer can be that the perimeter of the square is either:
4 * sqrt(2 * pi) or:
sqrt(32 * pi) or, if you need a decimal answer:
10.0265131 centimeters.
my problem was in visualizing the answer.
i finally got that right.
a graphical display of the circular coin and the square in the center of the coin would look like this:
you can see from the graph, that the length of each side of the square is 2 * 1.25331413732 = 2.506628275.
square this to get 2.506628275^2 = 6.283185307.
multiply this by pi / pi and it becomes 6.283185307 * (pi / pi) which is the same as (6.283185307 / pi) * pi which can be reduced to 2 * pi.
in other words, 6.283185307 is equivalent to 2 * pi, which is the area of the square that we had calculated to be earlier.