∏ The following note is P=C:2r=S:rr=CC:4S.
The third definition above has nothing to do with radius, so the questioner uses one of the first two to get a relationship with a third party to define radius.
In fact, the circular area can be understood as:
S= 1/2 * CR, take the center of the circle as the vertex, divide the circle into countless small triangles, and then quadrature and sum.
With this idea, the radius of any figure is certain.
R=2S/C, diameter D=4S/C, with obvious meaning and no gouge marks.
Similarly, for a sphere, it is divided into countless small cones and then summarized.
V= 1/3*SR，R=3V/S，D=6V/S。
This concept can also be extended to high-dimensional space.
Some friends think it is inappropriate to define the radius of a non-circular figure. In fact, the term radius is used in many places, such as the focal radius of ellipse, the curvature radius of curve, the convergence radius of power series, the radius of number set (neighborhood) and so on. Therefore, it is meaningful to define such a radius. In addition to the meaning I explained above, I believe we can describe its extreme average radius, and after further discussion, this definition will find its application, so this definition is not.
According to my statement, it is easy to understand why the radius obtained in this way is the radius of the inscribed circle. If you choose any regular polygon, its area is naturally the radius of the inscribed circle of S= 1/2*C*. Obviously, this also applies to other polygons circumscribed by any circle.
In addition, we can define a corresponding center on this basis, but it is not necessarily the center of mass.