正方形是几何学中一种具有特殊性质的四边形，它的四条边相等且四个角均为直角。圆是一个平面上离一个定点距离相等的点的集合，这个定点被称为圆心，距离被称为半径。现在我们来探讨正方形内部的一个圆以及它所阴影的部分。

在正方形内部画一个圆，首先要确定这个圆的位置。由于正方形的对角线相互垂直且长度相等，我们可以将圆的圆心放置在正方形的中心点，这样圆将与正方形相切于四个顶点。

接下来我们需要计算这个圆阴影部分的面积。圆阴影部分指的是正方形内部不被圆覆盖的区域。要计算这个面积，可以先计算整个正方形的面积，再减去圆的面积。

正方形的面积计算公式是边长的平方。假设正方形的边长为a，那么它的面积就是a的平方，即A = a^2。

圆的面积计算公式是π乘以半径的平方。假设圆的半径为r，那么它的面积就是πr^2，即C = πr^2。

所以圆阴影部分的面积可以表示为A - C。即阴影部分的面积等于正方形的面积减去圆的面积。用公式表示就是A - C = a^2 - πr^2。

这个式子可以进一步简化为 A - C = (a^2 - πr^2)。这里的a和r都是已知的数值，可以通过给定的问题来确定。

例如，假设正方形的边长为10厘米，圆的半径为5厘米。那么可以将这些数值代入公式中计算出阴影的面积。A - C = (10^2 - π5^2)。

在计算器上输入相应的数值进行计算后得到结果，最后转换成合适的单位。在本例中，可以得出阴影部分的面积为50π平方厘米。

In English:

A square is a quadrilateral in geometry with special properties, characterized by equal sides and right angles at each of the four corners. A circle, on the other hand, is a set of points on a plane equidistant from a fixed center point, with the distance referred to as the radius. Now let's explore a circle inside a square and the shaded area it creates.

To draw a circle inside a square, we first need to determine its position. Since the diagonals of a square are perpendicular and equal in length, we can place the circle's center at the center point of the square, causing the circle to touch the square at its four corners.

Next, we need to calculate the area of the shaded region created by the circle. The shaded region refers to the area inside the square that is not covered by the circle. To calculate this area, we can subtract the area of the circle from the total area of the square.

The formula to calculate the area of a square is the square of its side length. Let's assume the side length of the square is a, then its area is given by A = a^2.

The formula to calculate the area of a circle is π multiplied by the square of its radius. Assuming the radius of the circle is r, its area is given by C = πr^2.

Therefore, the area of the shaded region can be expressed as A - C, where A represents the area of the square and C represents the area of the circle. In equation form, we have A - C = a^2 - πr^2.

This equation can be further simplified as A - C = (a^2 - πr^2). Here, both a and r are known values that can be determined based on the given problem.

For example, let's assume the side length of the square is 10 cm and the radius of the circle is 5 cm. We can plug these values into the formula to calculate the area of the shaded region. A - C = (10^2 - π5^2).

After performing the calculations on a calculator, we obtain the result and convert it into appropriate units. In this example, the area of the shaded region is found to be 50π square centimeters.