I am not quite sure what your asking but I am guessing you
want to know the area of an arbitrary square inscribed in an arbitrary circle with
radius r.
First, you must note that the diagonal of a
square inscribed in a circle is equal to the diameter of that circle. Also, the radius
of a circle is half of it's diameter. So, d=2r and the diagonal of the square is also
2r.
Now, because we know that 2 adjacent sides of a square
along with its diameter forms an equilateral right triangle we can use Pythagorean
theorem, a^2+b^2=c^2, to find the length of each side of a square in terms of the radius
r. In this case both sides, a and b, are equal so lets just
say:
a=b=x.
so we rewrite the
theorem as
x^2+x^2=c^2.
we can
rewrite it further
as:
2(x^2)=c^2
and again since
c is equal to the diagonal of a square we can say c=2r and rewrite the theorem, once
more, as:
2(x^2)=(2r)^2.
so
now we solve for
x.
x^2=((2r)^2)/2.
x=sqrt((2r^2)/2).
since
the area of a square is the square of any of its equal sides we can now say that the
area of the square
is:
x^2=((2r)^2)/2.
Simplified,
x^2=(4r^2)/2.
Simplified once more,
x^2=2r^2.
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