Thursday, March 27, 2014

What is the answer for a square inscribed in a circle when the radius=R?

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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