Tikz pic angle around circle

Question

I am trying do draw an angle around an existing circle.

For example, I can draw 10 degree angle:

% Drawing circle
    \draw (0,0) circle [radius=1];

    % Drawing 10 degree angle sides
    \foreach \x in {0,10,...,360}
        {
            \ifthenelse{\x=10 \OR \x=0}{\draw (0,0) coordinate (O) -- ({cos(\x)*1.3},{sin(\x)*1.3}) coordinate (Deg\x)}{\draw[] (0,0) coordinate (O) -- ({cos(\x)},{sin(\x)}) coordinate (Deg\x)};
        }

    % Drawing 10 degree angle
    \pic["$10\degree$",draw=black, angle eccentricity=1.3, angle radius=2.5cm, ->] {angle=Deg0--O--Deg10};

I need to draw 360 degree angle. The problem is that it is drawing a "zero" angle instead of drawing a line around circle that starts at (0,0) and ends at (0,0).

% Drawing 360 degree angle
    \pic["$360\degree$", angle eccentricity=1.3, angle radius=2.5cm, ->] {angle=Deg0--O--Deg360};

EDIT:

I need to keep black circle and 10 degree angle and also add 360 degree angle:

Answer 1

A circle with radii at each 10 degrees can be drawn with simple means:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
  \def\Radius{2.5cm}
  \draw
    \foreach \a in {0, 10, ..., 350} {
      (0, 0) -- (\a:\Radius)
    }
    (0, 0) circle[radius=\Radius]
  ;
\end{tikzpicture}
\end{document}

Including the annotations:

\documentclass{article}
\usepackage{tikz}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}
  \def\Radius{2.5cm}
  \draw
    \foreach \a in {20, 30, ..., 350} {
      (0, 0) -- (\a:\Radius)
    }
    (0, 0) circle[radius=\Radius]
    (0, 0) -- (0:3cm) node[right] {\SI{360}{\degree}}
    (0, 0) -- (10:3cm)
  ;
  \draw[->] (0:2.85cm) -- (0:2.85cm);
\end{tikzpicture}
\end{document}

Next guess:

\documentclass{article}
\usepackage{tikz}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}
  \def\Radius{2.5cm}
  \draw[->, radius=\Radius]
    (0, 0) -- (0:3cm) node[right] {\SI{360}{\degree}}
    (0:\Radius)
    arc[start angle=0, end angle=180]
    arc[start angle=180, end angle=360]
  ;
\end{tikzpicture}
\end{document}

Combined version

\documentclass{article}
\usepackage{tikz}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}
  \def\Radius{2.5cm}
  \draw
    \foreach \a in {10, 20, ..., 350} {
      (0, 0) -- (\a:\Radius)
    }
    (0, 0) circle[radius=\Radius]
    %
    (0, 0) -- (0:3.75cm)
    (0, 0) -- (10:3.75cm)
    %
    (5:4cm) node {\SI{10}{\degree}}
    (-30:3.7cm) node {\SI{360}{\degree}}
  ;

  \def\Radius{3.5cm}
  \draw[->]
    (0:\Radius) arc[start angle=0, end angle=10, radius=\Radius]
  ;

  \def\Radius{3cm}
  \draw[->]
    (0:\Radius)
    arc[start angle=0, end angle=180, radius=\Radius]
    arc[start angle=180, end angle=360, radius=\Radius]
  ;
\end{tikzpicture}
\end{document}

Answer 2

Is this what you want?

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}
  \def\Radius{2.5cm}
  \draw
    \foreach \a in {10, 20, ..., 350} {
      (0, 0) -- (\a:\Radius)
    }
    (0, 0) circle[radius=\Radius]
    (0, 0) -- (0:3cm) node[right] {\SI{360}{\degree}}
  ;
  \draw[-LaTeX, red] (0:2.85cm) arc (0:360:2.85cm);
\end{tikzpicture}
\end{document}