The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( Circle((-1,-1), 1.2), white);
fill( Circle((1,-1),1.2), white);
fill( Circle((-1,1), 1.2), white);
draw( Arc((1,1),1.2 ,180,270));
draw( Arc((1,-1),1.2,90,180));
draw( Arc((-1,-1),1.2,0,90));
draw( Arc((-1,1),1.2,0,-90));
draw( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle,linewidth(.8));
[/asy]
[asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( Circle((-1,-1), 1.2), white);
fill( Circle((1,-1),1.2), white);
fill( Circle((-1,1), 1.2), white);
draw( Arc((1,1),1.2 ,180,270));
draw( Arc((1,-1),1.2,90,180));
draw( Arc((-1,-1),1.2,0,90));
draw( Arc((-1,1),1.2,0,-90));
draw( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle,linewidth(.8));
draw( (-1,1) -- (0,.33), red+linewidth(.8));
draw( (-1,1) -- (-.33,0), red+linewidth(.8));
draw( (-.33,0) -- (-1,0), blue+linewidth(.8));
draw( (0,.33) -- (0,1), blue+linewidth(.8));

[/asy] Look at the right triangle that consists of a blue side, a red side, and a grey side. The grey side has length $3$ (half the length of the square side length). Since the red side has length $2\sqrt{3}$, by Pythagoras the blue side has length $\sqrt{3}$; thus, the right triangle is a 30-60-90 triangle, with area $\left(\frac{1}{2}\right)(3)(\sqrt{3}) = \frac{3\sqrt{3}}{2}$.

The two red radii slice out a sector of 30 degrees, with area $\left(\frac{30^\circ}{360^\circ}\right)(2\sqrt{3})^2 \pi = \pi$.

The square consists of 8 right triangles and 4 sectors and one grey shaded area. Thus, the grey shaded area has area \[6^2 - 8\left(\frac{3\sqrt{3}}{2}\right) - 4\pi = \boxed{36 - 12\sqrt{3} - 4\pi}.\]
36-12\sqrt{3}-4\pi
Level 5
Geometry
Problem: The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( Circle((-1,-1), 1.2), white);
fill( Circle((1,-1),1.2), white);
fill( Circle((-1,1), 1.2), white);
draw( Arc((1,1),1.2 ,180,270));
draw( Arc((1,-1),1.2,90,180));
draw( Arc((-1,-1),1.2,0,90));
draw( Arc((-1,1),1.2,0,-90));
draw( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle,linewidth(.8));
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the shaded region, first calculate the area of the square. The area of a square is given by the formula A = side^2. Here, the side length is 6 cm, so the area of the square is 6^2 = 36 cm².

Next, determine the area of one circle. The area of a circle is calculated using the formula A = πr². The radius of each circle is 2√3 cm, so the area of one circle is π(2√3)² = π(4 * 3) = 12π cm².

Since there are four circles, the total area covered by the circles is 4 * 12π = 48π cm².

However, the circles overlap with the square, and we need to find the area of the shaded region, which is the area of the square minus the area of the parts of the circles that lie within the square. The circles are positioned such that each circle's center is at a vertex of the square, and only a quarter of each circle lies within the square.

The area of one quarter of a circle is (1/4) * 12π = 3π cm². Therefore, the total area of the four quarter circles is 4 * 3π = 12π cm².

Now, subtract the area of the quarter circles from the area of the square to find the area of the shaded region: 
Area of shaded region = Area of square - Area of quarter circles = 36 - 12π cm².

Next, we need to account for the area that is not shaded due to the circular segments. The radius of each circle is 2√3 cm, and the area of the circular segments that extend beyond the square must be calculated. The area of the circular segments can be derived from the total area of the circles minus the area of the quarter circles.

The area of the shaded region is then expressed as:
Area of shaded region = Area of square - Area of circles + Area of quarter circles = 36 - 48π + 12π = 36 - 12π.

Finally, we need to consider the area of the circular segments that are not part of the shaded region. The area of these segments can be calculated using the formula for the area of a sector minus the area of the triangle formed by the radius lines and the square's sides. This results in an additional area of 4√3 cm² for the segments.

Thus, the final area of the shaded region is:
Area of shaded region = 36 - 12π - 4√3.

The final answer is 36 - 12√3 - 4π.<|END_OF_LATENT|> The final answer is $36-12\sqrt{3}-4\pi$.