To make this easier, assume that $\displaystyle BX = 13$ and $\displaystyle BW=CX = 2$ - the reasoning generalizes. Then Square $\displaystyle ABCD$ has sidelength 15 and area $\displaystyle 15^{2}= 225$. The sidelength of Square $\displaystyle WXYZ$, each side being a hypotenuse of a right triangle with legs 2 and 13, is 

$\displaystyle \sqrt{2^{2}+13^{2}}= \sqrt{4+169} = \sqrt{173}$.

The square of this, 173, is the area of Square $\displaystyle WXYZ$.

The ratio is $\displaystyle 225:173$.

To make this easier, assume that $\displaystyle BX = 7$ and $\displaystyle BW = XC = 1$; the results generalize. 

Each side of Square $\displaystyle ABCD$ has length 8, so the area of Square $\displaystyle ABCD$ is 64. 

Each of the four right triangles has legs 7 and 1, so each has area $\displaystyle \frac{1}{2} \cdot 1 \cdot 7 = 3\frac{1}{2}$; Square $\displaystyle WXYZ$ has area four times this subtracted from the area of Square $\displaystyle ABCD$, or

$\displaystyle 64 - 4 \cdot 3 \frac{1}{2} = 50$.

The area of Square $\displaystyle WXYZ$ is

$\displaystyle \frac{50}{64} \times 100 \% = 78 \frac{1}{8} \%$

of that of Square $\displaystyle ABCD$.

Of the five choices, 80% comes closest.

The formula for the area of a circle is

$\displaystyle A = \pi r^{2}$.

If the area is 100, the radius is as follows:

$\displaystyle \pi r^{2} = 100$

$\displaystyle r^{2} = \frac{100}{\pi}$

$\displaystyle r = \sqrt{\frac{100}{\pi}}= \frac{10}{\sqrt{\pi}}$

The circle has circumference $\displaystyle 2 \pi$ times its radius, or

$\displaystyle C= \frac{10}{\sqrt{\pi}} \cdot 2 \pi = 20 \sqrt{\pi}$

This is also the perimeter of the square, so the sidelength of the square is one-fourth this, or

$\displaystyle S = \frac{1}{4} \cdot 20 \sqrt{\pi} = 5 \sqrt{\pi}$

The area of the square is the square of this, or 

$\displaystyle A'= \left (5 \sqrt{\pi} \right )^{2}= 25 \pi$

A circle with radius 8 has as its circumference $\displaystyle 2 \pi$ times this, or 

$\displaystyle 2 \pi \cdot 8 = 16 \pi$.

This is also the perimeter of the square, so the sidelength is one fourth of this, or 

$\displaystyle \frac{1}{4} \cdot 16 \pi = 4 \pi$.

The area is the square of this, or

$\displaystyle \left ( 4 \pi \right )^{2} = 16 \pi^{2}$.

The length of the hypotenuse of a right triangle with legs 8 and 12 can be determined using the Pythagorean Theorem:

$\displaystyle \sqrt{8^{2}+12^{2}} = \sqrt{64+144}= \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4\sqrt{13}$

Since this is also the perimeter of the square, its sidelength is one fourth of this, or

$\displaystyle \frac{ 4\sqrt{13} }{4} =\sqrt{13}$

The area of the square is the square of this sidelength, or 

$\displaystyle \left ( \sqrt{13} \right )^{2}= 13$

The perimeter of a square, and any shape for that matter, is found by adding up all the exterior sides. Since all sides are equal in a square, we can say: $\displaystyle P=x + x + x + x=4x$

where $\displaystyle x$ represents the length of a side

We can solve for the side length using the information provided:

$\displaystyle 4x=108$

$\displaystyle x=27$

The area of a square is found by squaring the side length: 

$\displaystyle A=x^2=27^2=729$

The length of one side of a square is the perimeter divided by 4:

$\displaystyle \frac{4 ^{d} }{4} = \frac{4 ^{d} }{4 ^{1}} = 4 ^{d-1}$

Square this to get the area:

$\displaystyle ( 4 ^{d-1} )^{2} = 4 ^{(d-1) \cdot 2 } = 4 ^{2d- 2 }$