Solution:
The diagram for the given problem is as follows:


Step 1: Find the length of the diagonal of the hexagon

The length of each side of the hexagon is 10 cm. Since it is a regular hexagon, all sides are equal.

Let's draw the radii of the circle from the center to each of the vertices of the hexagon as shown below:


Since the radii of the circle are also the perpendicular bisectors of the sides of the hexagon, we can form a right-angled triangle with the hypotenuse as 10 cm (length of each side of the hexagon) and one of the legs as 5 cm (half the length of each side of the hexagon).

Using the Pythagorean theorem, we can find the length of the other leg (which is also the radius of the circle):

$radius^2 = hypotenuse^2 - leg^2$
$radius^2 = 10^2 - 5^2$
$radius^2 = 75$
$radius = \sqrt{75}$
$radius = 5\sqrt{3}$

Step 2: Find the length of the side of the square

Since the square is inscribed inside the circle, its diagonal is equal to the diameter of the circle.

The diameter of the circle is twice the radius:

$diameter = 2 \times radius = 2 \times 5\sqrt{3} = 10\sqrt{3}$

Using the Pythagorean theorem, we can find the length of the side of the square (which is also the hypotenuse of a right-angled triangle with legs equal to the radius of the circle):

$side^2 = hypotenuse^2 - leg^2$
$side^2 = (10\sqrt{3})^2 - (5\sqrt{3})^2$
$side^2 = 300 - 75$
$side^2 = 225$
$side = 15$

Step 3: Find the area of the square

The area of the square is given by:

$area = side^2 = 15^2 = 225$

However, we need to express the area in square centimeters, since the length of the sides of the hexagon is given in centimeters.

Thus, the area of the square is 225 sq. cm.

Step 4: Verify the answer

We can verify the answer by calculating the area of the hexagon and checking if it is equal to the sum of the areas of the square and the circle.

The area of a regular hexagon with side length 'a' is given by:

$area = \frac{3\sqrt{3}}{2}a^2$

Substituting 'a' as 10 cm, we get:

$area = \frac{3\sqrt{3}}{2} \times 10^2 = 300\sqrt{3}$

The area of the circle with radius 5√3 cm is given by:

$area = \pi r^2 = \pi (5\sqrt{3