正方形是几何学中一种特殊的多边形，具有四个相等的边和四个相等的角。圆则是一个平面上所有离圆心的距离都相等的点的集合。如果我们将一个圆放置在正方形内部，我们可以推导出阴影的面积。

首先，让我们考虑正方形的边长为a个单位。这意味着正方形的周长为4a。由于正方形的四个边是相等的，所以每条边的长度都是a。

现在，让我们将一个圆放置在正方形的内部，使得圆与正方形的四个边界相切。这意味着圆的直径等于正方形的边长a。

圆的直径是圆上任意两点之间的最大距离，也是圆的最长线段。由于圆与正方形的四个边界相切，所以圆的直径等于正方形的边长，即2r = a（其中r为圆的半径）。

根据上述关系，我们可以计算出圆的半径为r = a / 2。因此，圆的半径为正方形边长的一半。

接下来，我们需要计算阴影的面积。阴影的面积是指圆没有覆盖到的正方形的面积。

正方形的面积可以通过边长的平方来计算，即A = a^2。

圆的面积可以通过半径的平方乘以π（圆周率）来计算，即A' = π * r^2。

阴影的面积可以通过正方形的面积减去圆的面积来计算，即S = A - A' = a^2 - π * r^2。

将r替换为a / 2，我们得到S = a^2 - π * (a / 2)^2。

简化上述表达式，我们得到S = a^2 - π * (a^2 / 4)。

进一步简化，我们得到S = a^2 - (π / 4) * a^2。

因此，阴影的面积可以表示为S = (1 - π / 4) * a^2。

最后，让我们计算阴影的面积。假设正方形的边长为5个单位，则阴影的面积为S = (1 - π / 4) * 5^2。

通过计算，我们可以得出阴影的面积约为12.57平方单位。

总结起来，当一个圆放置在正方形内部，与正方形的四个边界相切时，阴影的面积可以表示为(1 - π / 4) * a^2，其中a为正方形的边长。根据具体的边长数值，我们可以计算出阴影的面积。

English Translation:

A square is a special polygon in geometry, with four equal sides and four equal angles. On the other hand, a circle is a set of points on a plane that are equidistant from a center point. If we place a circle inside a square, we can calculate the area of the shadow that it creates.

Firstly, let's consider a square with side length a units. This means that the perimeter of the square is 4a. Since all four sides of the square are equal, each side has a length of a.

Now, let's place a circle inside the square, such that the circle is tangent to all four boundaries of the square. This implies that the diameter of the circle is equal to the side length of the square, a.

The diameter of a circle is the longest line segment that can be drawn within the circle, connecting any two points on its circumference. Due to the circle being tangent to the four boundaries of the square, the diameter of the circle is equal to the side length of the square, i.e., 2r = a (where r is the radius of the circle).

From this relationship, we can deduce that the radius of the circle is r = a / 2. Therefore, the radius of the circle is half of the side length of the square.

Next, we need to calculate the area of the shadow. The shadow represents the area of the square that is not covered by the circle.

The area of the square can be calculated by squaring its side length, i.e., A = a^2.

The area of the circle can be calculated by multiplying the square of its radius by π (pi), i.e., A' = π * r^2.

The area of the shadow is obtained by subtracting the area of the circle from the area of the square, i.e., S = A - A' = a^2 - π * r^2.

Substituting r with a / 2, we have S = a^2 - π * (a / 2)^2.

Simplifying the above expression, we obtain S = a^2 - π * (a^2 / 4).

Further simplifying, we have S = a^2 - (π / 4) * a^2.

Therefore, the area of the shadow can be represented as S = (1 - π / 4) * a^2.

Finally, let's calculate the area of the shadow. Assuming the side length of the square is 5 units, the area of the shadow is S = (1 - π / 4) * 5^2.

Through calculation, we find that the area of the shadow is approximately 12.57 square units.

In conclusion, when a circle is placed inside a square and is tangent to all four boundaries of the square, the area of the shadow can be expressed as (1 - π / 4) * a^2, where a is the side length of the square. Using the specific value of the side length, we can calculate the area of the shadow.