regular polygon

A regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length)

May be convex, star, or skew

In the limit of an $n$-gon as $n\to \mathrm{\infty }$,
if perimeter/area is fixed, sequence approximates a circle
if side length is fixed, sequence approximates regular apeirogon

A regular $n$-sided polygon ($n$-gon):

• has rotational symmetry order $n$
• has all vertices lying on the circumscribed circle
• has an inscribed circle or incircle tangent to every side at its midpoint
• symmetry group is dihedral group $D_n$ (of order $2n$)
  • rotations in $C_n$
  • reflections in $n$ axes

apothem: distance from the center to any side of a regular polygon

\documentclass[tikz]{standalone}
%
\begin{document}
	\begin{tikzpicture}[scale=3]
	    % \draw (0,0) circle (1);
	    %
	    % Specify number of sides/vertices:
	    \def\n{9}
	    %
	    % Place a northernmost reference node (for edge calculation):
	    \node (v-0) at (90:1) {};
	    % Place nodes for vertices and edges (iterate n times):
	    \foreach \k in {1,...,\n} {\
		    % Place a node at vertex:
	        \node (v-\k) at (\k*360/\n+90:1) {$v\k$};
	        % Index previous to c:
	        \pgfmathsetmacro\c{int(\k-1)}
	        % Place a node on edge between current and previous:
	        \path (v-\k.center) -- (v-\c.center) node[midway](e-\k){$e\k$};
	    }
	    % Draw a cycle from v-1 to v-2 to ... to v-n:
	    \draw[very thick] (v-1.center) foreach \x in {2,...,\n}{-- (v-\x.center)} -- cycle;
	    % Note that v-0 is only used for placing e-1 and is incident with v-n
	\end{tikzpicture}
\end{document}