I think you would like to use a node, in this way you can reference the point afterward.
What you need to set the exact radius is inner sep=0pt, as explained here.
I'll show you the difference in this simple example. In the first circle, the center node has inner sep=0pt; in the second circle, the center node has the standard inner sep:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}
\node[draw,circle, minimum size=3cm, inner sep=0pt] (A) {};
\node[draw,circle, fill=black, inner sep=0pt, minimum width=1pt] (mycenter) at (A.center) {};
\node[draw,circle, minimum size=3cm, inner sep=0pt, below = of A] (B) {};
\node[draw,circle, fill=black, minimum width=1pt] (mycenter) at (B.center) {};
\end{tikzpicture}
\end{document}

Prof. van Duck explained this feature in his article The Morse code of TikZ, see Section 3.1.
\fill (0,0) circle[radius=0.5pt];? – Dec 14 '19 at 05:47\node[draw,circle,minimum size=1pt,inner sep=0pt,fill=blue!20] at (0,0){};. Useful to scale a picture without getting big points (small). – Alain Matthes Dec 18 '19 at 13:19\drawcommand. I have the same problem: How to force the\drawcommand to stroke the sigle points, eg. by\draw (0,0) -- (1,1) (1,0). Here is my problem: link – forrest Jan 11 '23 at 17:50