正方形里有个圆，这是一个颇为有趣的几何问题。我们将探索这个问题并求解阴影部分的面积。首先，让我们来了解一下正方形和圆的特性。

正方形是一种具有四个相等边长和四个直角的四边形。它的边界是由四条相互垂直的线段组成。正方形在几何学中具有许多重要的特性，比如对角线相等、内角和为360度等。

圆是一个特殊的椭圆，其每个点到圆心的距离都相等。圆由一个中心点和半径组成，其中半径是从中心点到圆上任意一点的距离。圆在许多领域中都具有广泛应用，比如建筑、机械工程和艺术设计等。

现在，我们回到问题本身。正方形里的圆可以通过将圆心放置在正方形的中心，并使圆与正方形的四个角点相切来实现。这样做可以确保圆与正方形的边界相切，并且圆会刚好完全位于正方形的内部。

为了求解阴影部分的面积，我们需要计算正方形的面积和圆的面积，并从正方形的面积中减去圆的面积。正方形的面积计算公式是边长的平方，而圆的面积计算公式是π乘以半径的平方。

假设正方形的边长为L，圆的半径为R。那么正方形的面积等于L的平方，即L^2；圆的面积等于π乘以R的平方，即πR^2。因此，阴影部分的面积等于L^2减去πR^2。

现在，我们可以进一步简化求解过程。由于正方形的四个边长相等，所以L可以表示为L=2R。将L代入面积计算公式中，可以得到阴影部分的面积等于4R^2减去πR^2，即(4-π)R^2。

最后，我们得出了阴影部分的面积公式，即A = (4-π)R^2。这个公式告诉我们，在给定正方形和圆的条件下，阴影部分的面积只取决于圆的半径。通过改变圆的半径，我们可以调整阴影部分的面积大小。

In a square with a circle, the shadowed area is an interesting geometric problem. We will explore this problem and calculate the area of the shadowed region. First, let's understand the properties of a square and a circle.

A square is a quadrilateral with four equal sides and four right angles. Its boundaries consist of four perpendicular line segments. The square has several important properties in geometry, such as equal diagonals and the sum of interior angles being 360 degrees.

A circle is a special case of an ellipse where every point on the circle is equidistant from the center. It is defined by a center point and a radius, which is the distance from the center to any point on the circumference. Circles find wide applications in various fields, such as architecture, mechanical engineering, and art design.

Now, let's get back to the problem at hand. The circle within the square can be achieved by placing the center of the circle at the center of the square and making the circle tangent to the four corner points of the square. This ensures that the circle is tangent to the boundaries of the square and completely enclosed within it.

To calculate the area of the shadowed region, we need to compute the areas of both the square and the circle and subtract the area of the circle from the area of the square. The area of a square is given by the formula side length squared, while the area of a circle is given by π times the radius squared.

Let's assume the side length of the square is L, and the radius of the circle is R. The area of the square is then L^2, and the area of the circle is πR^2. Therefore, the area of the shadowed region is obtained by subtracting the area of the circle from the area of the square, which gives (L^2 - πR^2).

Now, we can further simplify the solution. Since all sides of the square are equal, we can express L as L = 2R. Substituting L into the area formula, we get the area of the shadowed region as (4R^2 - πR^2), which can be further simplified to (4-π)R^2.

In conclusion, we have derived the formula for the area of the shadowed region, A = (4-π)R^2. This formula tells us that, given the square and the circle, the area of the shadowed region depends solely on the radius of the circle. By varying the radius, we can adjust the size of the shadowed area.