The diagram shows eight congruent squares inside a circle. Every shaded square has one vertex on the circle. What is the ratio of the shaded area to the area of the circle? Express your answer as a decimal to the nearest hundredth.
[asy]
pair A,O,B,C,D,EE,F,G,H;
A = (0,0);
B = (17.68,8.68);
C = (10,17.68);
D = (-8.68, 10);
EE = (-17.68,-8.68);
F = (-10,-17.68);
G = (8.68,-10);
H = (6.34, 0);
draw(A--B--C--D--EE--F--G--H--cycle,dashed);
draw(Circle((0,0),20));
fill(A--B--H--cycle,gray);
fill(A--H--G--F--cycle,gray);
fill(A--F--EE--D--cycle,gray);
fill(A--D--C--B--cycle,gray);
draw((-40,0)--(40,0),linewidth(0.7),Arrows(6));
draw((0,-40)--(0,40),linewidth(0.7),Arrows(6));
label("$x$",(40,0),SE);
label("$y$",(0,40),NW);
dot((-10,0));dot((0,0));dot((10,0));dot((20,0));
label("$-2$",(0,0),NW);
label("$-1$",(10,0),NW);
label("$0$",(20,0),NW);
label("$1$",(-10,0),NW);
[/asy]
We label the points as shown. The radius of the circle is labelled 20. Let the sidelength of each square be $s$. From the tangent part of the diagram, $2s = 20$, so $s = 10$.
The area of each square is then $s^2 = 10^2 = 100$. At this point, let's compute the area of the shaded region. We can clearly see that it consists of 8 congruent triangles. Each triangle has base $10$, so we only need to find the height.
[asy]
pair A,O,B,C,D,EE,F,G,H;
O = (0,0);
B = (17.68,8.68);
C = (10,17.68);
D = (-8.68, 10);
EE = (-17.68,-8.68);
F = (-10,-17.68);
G = (8.68,-10);
H = (6.34, 0);
draw(Arc(O,20,0,180));
draw(O--B--C--D--F--G--H,dashed);
draw(O--C--H--cycle);
draw(rightanglemark(O,H,C,40));
draw((-40,0)--(40,0),linewidth(0.7),Arrows(6));
draw((0,-40)--(0,40),linewidth(0.7),Arrows(6));
label("$x$",(40,0),SE);
label("$y$",(0,40),NW);
dot((-10,0));dot((0,0));dot((10,0));dot((20,0));
label("$-2$",(0,0),NW);
label("$-1$",(10,0),NW);
label("$0$",(20,0),NW);
label("$1$",(-10,0),NW);
[/asy]
We compute the area of each triangle by finding the area of the rectangle (since it has two right angles) that it is a part of and then divide by $2$. We know that the length of the rectangle is $10$, so we need to find the width and height. Using the Pythagorean Theorem, we know that $x^2 + y^2 = 20^2 = 400$, so $y = \pm \sqrt{400 - x^2}$. Looking at our diagram suggests that $y$ is positive, so $y = \sqrt{400 - x^2}$.
The area of our rectangle is then $10 \cdot \sqrt{400 - x^2}$. Now we must find the height of each triangle. Recall that the height of a right triangle is the side opposite the right angle. So, we have $10-2s = 10 - 2 \cdot 10 = -10$. But our answer must be positive! Don't forget that we subtracted positive $2s$. To get a positive value, let's just take the absolute value of $10 - 2s$, which gives us $|-10| = 10$.
To find the area of each triangle and hence the shaded area, we multiply the base by the height, and divide by 2. So $\frac{10^2 \cdot \sqrt{400 - x^2}}{2} = 50 \sqrt{400 - x^2}$.
Now, let's divide by the area of the circle. The area of the circle is $\pi \cdot 20^2 = 400 \pi$. Thus, the ratio is $\frac{50 \sqrt{400 - x^2}}{400 \pi}$. Furthermore, the ratio is the same for all $x$ because all 8 shaded triangles are congruent, and they comprise the entire shaded area. Hence we can use any value of $x$ in our formula. Let's use $x=2$ because it gives us a clean right triangle.
We plug in $x=2$ to get $\frac{50 \cdot \sqrt{400 - 2^2}}{400 \pi} = \frac{50 \cdot \sqrt{394}}{400 \pi}$. We rationalize the denominator of the square root expression:
$$\frac{50 \cdot \sqrt{394}}{400 \pi} =\frac{50 \cdot \sqrt{394}(20 \sqrt{394})}{400 \pi(20 \sqrt{394})}=\frac{50 \cdot 20 \cdot 394}{400\pi \cdot 20 \sqrt{394}}=\boxed{\frac{197}{40\sqrt{394}\pi}}.$$
To find the ratio of the shaded area to the area of the circle, we need to determine the area of one shaded square and the area of the circle.
Let's assume the side length of one square is "s."
The area of one shaded square is given by s^2.
Since all the squares are congruent, the area of any shaded square will be the same as any other shaded square.
Now, let's consider the circle.
The formula for the area of a circle is A = πr^2, where r is the radius.
In this case, the radius of the circle is equal to half the length of the square's side, so r = s/2.
Substituting the value of r into the formula, we have A = π(s/2)^2 = πs^2/4.
To find the ratio of the shaded area to the area of the circle, we divide the area of one shaded square by the area of the circle.
(shaded area) / (circle area) = (s^2) / (πs^2/4) = 4/π.
Using a calculator, we can evaluate this ratio to the nearest hundredth: 4/π ≈ 1.27.
Therefore, the ratio of the shaded area to the area of the circle is approximately 1.27.