You need to remember that the vertices of the square lie on the circumference of circle and the center of circle is also the midpoint of diagonals of the square.

Notice that diagonals of the inscribed square are also diameters of circle such that:

`d = 2R`

You need to remember that diagonal of the square is hypotenuse in rigth angle triangle that has the legs equal to x.

Using Pythagorean theorem yields:

`(2R)^2 = x^2 + x^2 =gt 4R^2 = 2x^2 =gt 2R^2 = x^2`

`x = Rsqrt2`

**Hence, evaluating the dimensions of the square under given conditions, yields that the length of each side of inscribed square is `x = Rsqrt2 ` and the length of diagonal of the square is `d = 2R` .**

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.

**since the square is inscribed in the circle, then all 4 points of the square lie on the circle. since the diagonals of a square are equal to each other, then each diagonal must be a diameter of the circle and they must pass through the center of the circle. since the area of the square is 64 square inches, then each side of the square musts be 8 inches since 8*8 = 64 since each side of the square is 8 inches, then the diagonal of the square must be . since the diagonal of the square is also the diameter of the circle, this means that the radius of the circle must be 1/2 * 11.3137085 = 5.65685425. this means that the area of the circle is pi*(5.65685425)^2 = 100.5309649. can't go any further with this because i don't know what the shaded region is. if the shaded region is the area of the circle outside the square, then the area of the shaded region is the area of the circle minus the area of the square. if the shaded region is the area of the circle outside the square, but only one of the 4 sections, then the area of that shaded region is one fourth times the (area of the circle minus the area of the square). hopefully this will help you get started at least.**