Description of Polygons

Polygons can best be described as three or more 
points joined by the same number of line segments.  The 
connecting line segments are known as sides of a polygon.
Whereas, the name given to the points, endpoint, of a 
polygon are called vertices [singular-vertex].

Initialization:

Clear[vR,n,poly,ppoly]
Off[General::spell1]
Off[General::spell]

vR[n_]:=
 Table[{Cos[t],Sin[t]},{t,0,2Pi, 2 Pi/n}];
	
poly[n_]:=ListPlot[vR[n],PlotJoined->True,
	AspectRatio -> Automatic,
	Axes->None,
	DisplayFunction->Identity];
	
pts[n_] := Map[Point,vR[n]];

ppoly[n_,k__:0]:=

Show[poly[n],

Table[Graphics[{If[k===1,RGBColor[0,0,1],RGBColor[0,0,0]],
Line[{vR[n][[k]],vR[n][[i]]}]}],
{i,3,Length[vR[n]]-2}],

Graphics[{PointSize[.04],RGBColor[1,0,0],pts[n]}],

                AspectRatio->1,
	            DisplayFunction->$DisplayFunction];

Polygons with points/sides ranging from three to ten 
have definite names.  The names of such polygons are 
identified as follows:

3 points/sided - Triangle

ppoly[3];

4 points/sided - Quadralateral

ppoly[4];

5 points/sided - Pentagon

ppoly[5];

6 points/sided - Hexagon

ppoly[6];

7 points/sided - Heptagon

ppoly[7]

8 points/sided - Octogon

ppoly[8]

9 points/sided - Nonagon

ppoly[9];

10 points/sided - Decagon

ppoly[10];

Polygons with more than ten points/sides are known 
according to the number of points/sides represented
in the geometric design.   For example, a polygon with 
eleven (11) points/sides is called an 11 - gon.

What name would be given to a polygon with 15 points/
	sides?

ppoly[15];

What name would be given to a polygon with 13 points/
	sides?

ppoly[13];

How many vertices/line segments are identified in the 
following polygons?

ppoly[4];

A diagonal of a polygon is a line segment.   The purpose of this line segment is to join two points, vertices.   The line segments known as diagonals can not join the sides of the geometric design.   Most polygons can have more than one diagonal drawn from one vertex. How many diagonals can be drawn fron a given vertex in the following polygons:

ppoly[5,1];

How many different diagonals does an N- points/sided polygon have?

Expansion:
How many different line segments connect the eight (8) vertices in the figure below?

allsegments[n_] :=
Show[Graphics[{ RGBColor[1.,0.,1.],
Table[Line[{vR[n][[i]],vR[n][[j]]}],
{i,1,n},{j,1,n}]}],AspectRatio->1]

allsegments[7]