Polygon – a plane figure that is formed by three or more segments such that

1) each segment intersects exactly two other segments at an endpoint and

2) no two segments with a common endpoint are collinear.

Polygon – From Greek: polys “many” + gonia “angle”

Polygons are a number of coplanar line segments, each connected end to end to form a closed shape.

Polygons are one of the most all-encompassing shapes in geometry. From the simple triangle up through squares, rectangles, trapezoids, to dodecagons and beyond.

This is polygon *ABCDE*. or polygon *AEDCB*. or polygon *BCDEA*. or polygon *BAEDC*. or yawn. A polygon is named by picking one of the vertices and proceeding either clockwise or counterclockwise. No rearranging letters.

Points *A*, *B*, *C*, *D*, and *E* are this polygon’s vertices.

∠*A*, ∠*B*, ∠*C*, ∠*D*, and ∠*E* are this polygon’s angles.

*AB*, *BC*, *CD*, *DE*, and *AE* are this polygon’s sides.

.

Diagonal – a segment that connects two non-consecutive vertices

*AC* and *AD* are diagonals of this polygon starting at vertex *A*.

A diagonal of a polygon is a line segment joining two vertices. From any given vertex, there is no diagonal to the vertex on either side of it, since that would lay on top of a side. Also, there is obviously no diagonal from a vertex back to itself.

The formula n(n – 3)/2 where n is the number of sides in the polygon, gives the number of distinct diagonals a polygon can have.

A pentagon will have 5(5 – 3)/2 = 5 distinct diagonals

A triangle will have 3(3 – 3)/2 = 0 diagonals

…..…..

.

Convex polygon – a polygon with all diagonals in the interior of the polygon

If a polygon is not convex, it is called concave or non-convex

.

A polygon is classified or named according to the number of its sides

3 sides triangle

4 sides quadrilateral

5 sides pentagon

6 sides hexagon

7 sides heptagon

8 sides octagon

9 sides nonagon

10 sides decagon

.

12 sides dodecagon

42 sides tetracontakaidigon

672 sides hexahectaheptacontakaidigon

There are some that wish to name every possible polygon, but there seems little point in doing so. Beyond about 10 sides, most people call them an “*n*-gon”. For example a 15-gon has 15 sides.

*n* sides *n* – gon

25 sides 25 – gon

The above examples were first a simple convex polygon and second a regular polygon. Notice how as another side was added, the polygons became more circular.

.

Equilateral polygon – a polygon with all its sides all congruent.

Equiangular polygon – a polygon with all its angles all congruent.

.

Triangles are the only polygons that are automatic equiangular if they are equilateral and vice versa.

.

Regular polygon – a polygon that is both equilateral and equiangular

.

Regular polygons tend to look like someone tried to make a circle out of some straight lines. In fact, if you have a polygon with very many sides, it looks a lot like a circle from a distance. The sides and vertices are evenly spread around a central point, and regular polygons are convex – all the vertices point ‘outwards’.

.

…. Regular 20-gon – The more sides, a regular polygon looks more and more like a circle.

…..Regular 40-gon – Or is this a circle? No, its a regular 40-gon.