Convert image into TikZ

Question

I want to convert this image into TikZ.

I already have the following code, but I don't know how to put the numbers at the vertices and how to put that red bar inbetween 2 and 3. Could someone please explain how to do this?

\documentclass{article}

\usepackage{xcolor}

\definecolor{navyblue}{rgb}{0.0, 0.0, 0.5}

\usepackage{
  pst-poly,
  pstricks-add
}
\usepackage[
%  locale = DE,
  round-mode = places,
  round-precision = 2
]{siunitx}

\ExplSyntaxOn
  \cs_new_eq:NN
    \calc
  \fp_eval:n
\ExplSyntaxOff

% calculations
\newcommand*\Angle{\calc{360/\sides}}
\newcommand*\sidelength{\calc{2*\radius*sin(pi/\sides)}}
\newcommand*\radiusI{\calc{\radius*cos(pi/\sides)}}
\newcommand*\areaI{\calc{pi*\radiusI^2}}%        not used
\newcommand*\areaC{\calc{pi*\radius^2}}%         not used
\newcommand*\areaRatio{\calc{cos(pi/\sides)^2}}% not used

\psset{dimen = middel}

\begin{document}

% constants
\newcommand*\sides{5}
\newcommand*\radius{3.5}

\begin{center}
\begin{pspicture}(-\radius,-\radius)(\radius,\radius)
  % centre
  \pnode(0,0){C}
  % regular polygon with dots at corners
  \rput(C){%
    \PstPolygon[
    PolyNbSides = \sides,
    unit = \radius,
    PolyRotation=18,
    linecolor = navyblue
    ]
  }
 {\psset{linecolor = gray}
  % circumscribed circle
  \pscircle(C){\radius}}

\end{pspicture}
\end{center}

\bigskip

\end{document}

Answer 1

A short code with pst-poly. I had to introduce a new ‘vertex’ counter, because the option PolyName=… allows to have numbered vertices, but their order is not the order required by the O.P.:

\documentclass[svgnames]{article}

\usepackage{pst-poly}
\usepackage{auto-pst-pdf}
\usepackage{etoolbox}
\colorlet{PolygonColor}{Thistle}
\colorlet{PolygonColor1}{red}
\colorlet{VertexColor}{NavyBlue}
\colorlet{LabelColor}{OrangeRed!80!}

\begin{document}

\psset{unit=3.5cm, dimen=middle, linejoin=1, dotsize=3.5pt, arrowinset=0.15, shortput=nab}
\begin{pspicture}(-1,-1)(1,1)
    % centre
    \pnode(0,0){C}
    % circumscribed circle
    \pscircle[linecolor =LightSlateGray!60](C){1}
    % regular polygon with dots at corners
    \psset{linecolor=PolygonColor}
    \newcounter{vertex}
    \setcounter{vertex}{4}
    \providecommand{\PstPolygonNode}{\rput * { * 0}(1.09;\INode){\addtocounter{vertex}{-1}\ifnumequal{\value{vertex}}{0}{\setcounter{vertex}{5}}{}\color{LabelColor}\thevertex}\psdots[linecolor =VertexColor](1;\INode)}
    \rput(0,0){\PstPentagon[PolyName=A, linecolor=white] }
    \psset{arrows=<-, arrowsize =2.5pt 2, nodesepA=1.7pt, nodesepB =1.75pt, linecolor=PolygonColor, linewidth=1.2pt}
    \ncline{A1}{A2}\ncline{A2}{A3}_{$\tau$}\ncline{A3}{A4}\ncline{A4}{A5}\ncline{A5}{A1}
    \ncline[offset=4pt, linecolor=PolygonColor1, arrows=<->]{A1}{A2}^{$ \sigma$}
\end{pspicture}

\end{document}

Answer 2

Here's a TikZ solution. It uses a foreach loop to draw the edges. The values for the loop ({2,1,a5,B,$\alpha$}) are used as labels, starting at the top and then clockwise. Furthermore count=\c tracks the current corner, and coordinates are created in each one (called c1, c2 and so on) so that you can access them easily afterwards. The markings decoration is used to place the arrow tips along the polygon, as there are five, each side represents 20% (or 0.2) of the whole path. Due to rounding inaccuracies, the placement of the arrows might be bad (for instance if the radius \r equals 1), to circumvent this one could place the tips more on the edges (easy) or change to another arrow placing facility (slightly less easy).

Code

\documentclass[tikz, border=2mm]{standalone}
\usetikzlibrary{decorations.markings}

\begin{document}

\begin{tikzpicture}
    \pgfmathsetmacro{\r}{3}
    \draw circle (\r);
    \draw[decoration={markings, mark=between positions 0.2 and 1 step 0.1999 with \arrow{latex}}, postaction=decorate] (162:\r)
    \foreach \l [count=\c] in {2,1,a5,B,$\alpha$}
    {   -- (162-\c*72:\r) coordinate (c\c) node at (162-\c*72:\r+0.3) {\l}
    };
    \draw[red, thick, -latex] (c3) -- (c5);
\end{tikzpicture}

\end{document}

Output

Answer 3

With tikz

\documentclass{article}
\usepackage{xcolor}
\usepackage{tikz}

\definecolor{navyblue}{rgb}{0.0, 0.0, 0.5}

% calculations
\newcommand*\Angle{\calc{360/\sides}}

\begin{document}

% constants
\newcommand*\sides{5}
\newcommand*\radius{3.5}

\begin{center}

\begin{tikzpicture}[>=stealth]
\draw[gray,thick](0,0)circle(\radius);
\foreach \i in {1,...,5}{\node[fill,circle,inner sep=2pt,label=90-(\i-2)*\Angle:$\i$](\i) at ({90-(\i-2)*\Angle}:\radius){};}

\foreach \i/\j in {1/2,2/3,3/4,4/5,5/1}{
\draw[thick,navyblue,->](\i)--(\j);}

\draw[red,<->](2)--(3)node[midway,below]{r};

\end{tikzpicture}
\end{center}

\end{document}

Update

To change numbers at the vertices you can modify variables inside \foreach like this \foreach \i/\j in {1/2,2/4,3/5,4/1,5/3}{\node[fill,circle,inner sep=2pt,label=90-(\i-2)*\Angle:$\j$](\i) at ({90-(\i-2)*\Angle}:\radius){};} to obtain numbers arranged as follows 2,4,5,1,3.