# What is a hexagon?

Having a slight parenting anxiety attack and I hate teaching my son something incorrect.

Wiktionary tells me that a Hexagon is a polygon with $6$ sides and $6$ angles.

Why the $6$ angle requirement? This has me confused.

Would the shape below be also considered a hexagon?

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Also, the angles should be mentioned because of the literal translation of ἑξάγωνον "six angles". On the other hand every polygon has the same number of sides and vertices - and angles, unless you allow degenerate cases with e.g. 180° angles. –  Hagen von Eitzen Oct 7 '12 at 9:03
It sounds like you experienced a slight case of "prototype phenomenon". Many people are only accustomed to seeing convex polygons, but what you have drawn is a perfectly good (nonconvex) hexagon. –  rschwieb Oct 7 '12 at 13:47
Hexagon, is produced by a greek word. Hexa= SIX(6) gon = angle . And as you can see , the shape above has 6 angles ,so it is a hexagon. –  george mano Oct 7 '12 at 17:57
I'm counting six angles in the figure. –  Christoffer Hammarström Oct 7 '12 at 21:17
I would like to say a big thanks to the Math community for embracing this question. I want my son (5yo) to love learning so I spend time with him doing math, reading and drawing. The book said "Colour the hexagons" and provided a large range of shapes including the one above. Thank you for helping us both to learn. –  xiaohouzi79 Oct 7 '12 at 23:21

Yes, it would still be considered a hexagon. The reason why we require "$6$ angles" is probably just because you don't want the $6$ lines to cross and create "more than $6$ angles". The hexagons you are probably thinking of (the one with all the angles and sides equal) would be regular hexagons.

Hope that helps.

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Just for being complete and accurate - "the one with all the sides and angles equal" should be the right phrase here. –  Shashank Sawant Oct 7 '12 at 15:36
@Shashank : Of course. It was 4 am when I wrote this, didn't think much about this kind of detail but it's true =) –  Patrick Da Silva Oct 7 '12 at 16:28

Yes. It also has six angles, but one of them is greater than $180^{\circ}$.

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Interesting to speculate about whether Euclid would have considered this a hexagon. Probably not, since he didn't believe in angles greater than 180 degrees. –  Ben Crowell Oct 8 '12 at 0:04

A hexagon is a simple closed polygon $P$ (presumably in the plane) with $6$ sides. Simple means that the sides are arranged in cyclic order, that two subsequent sides have exactly one point, called a vertex, in common, and that otherwise the sides don't intersect. Maybe you want to include the condition that subsequent sides should not be parallel. Jordan's curve theorem then guarantees that $P$ has a well defined interior (an "open hexagon") and an exterior which extends to infinity.

The polygon in your figure is a hexagon according to this definition. If you want to exclude it you would have to explicitly require that the considered hexagon be convex. (Maybe you can use this opportunity to make your son familiar with the notion of convexity $\ldots$)

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Nice answer, I've learned a new thing, what a vertex is :) –  Marian Zburlea Oct 7 '12 at 12:00
If the subsequent sides were parallel they would not share only one vertex so that condition is unnecessary. –  vakio Oct 8 '12 at 8:07
@vakio: Sides in the above definition are segments $[A_{k-1},A_k]$, not infinite lines. Consider an equilateral triangle from which a tiny triangle at the base has been removed. This hexagon has two sides lying on the same line. –  Christian Blatter Oct 8 '12 at 9:08

(This was way too long for a comment on @Joseph's answer.)

Branko Grünbaum was my graduate advisor, so I couldn't help but to adopt a very broad view of polygons and polyhedra. To me,

A "hexagon" consists of six not-necessarily-distinct points (the "vertices"), connected, in order, by six possibly-zero-length line segments (the "edges").

(The "edges" don't even really need to be straight line segments, but can be curves. For purposes of this discussion, my edges are straight.) The definition encompasses familiar convex and non-convex figures, ones whose edges meet or cross at points other than their vertices (although the crossing-points aren't considered new vertices), ones that multiply-trace a simpler figure (say, by going around a triangle twice), the one in which all vertices coincide and you can't see any edges (I call this the "dot"), and cases where the vertices aren't even confined to a plane.

Here's a figure —featuring "15-gons"— from my note, "Spectral Realizations of Graphs", that examines highly-symmetric versions of these broadly-defined polygons and polyhedra. Each picture contains a polygon defined by 15 vertices joined by 15 edges, but the edges are allowed to cross and the vertices are allowed to coincide. (The polygon in (a) is the lonely black "dot" at the far right of the ghostly circle of gray reference points.)

(The math of the note may be a little intimidating, but the bulk of the content comprises zillions of pictures that everyone can enjoy.)

An advantage of this broader (and yet not broadest!) view is that it allows every member of the "hexagon" family to be deformed into any other by simply moving vertices around, not having to worry about edges crossing or collapsing. This freedom leads to some remarkable mathematics involving "adding" figures.

Below is a figure from my note "An Extension of a Theorem of Barlotti" that deals with this notion of addition. The figure indicates how vertices from the regular pentagons combine to give a vertex of the "flatter" pentagon.

One needs the very broad view of polygons here, because the definition of "addition" applies vertex-by-vertex, with no concern for how results of each step relate to one another; it's possible in general, for instance, to jiggle the pentagons being added in such a way that the resulting pentagon has crossed or collapsed edges.

Ultimately, one can combine the results of my two notes and declare, for instance, that

Any hexagon —as weirdly deformed (but straight-edged), self-intersecting, edge-collapsed, or non-planar as you like— is the "sum" of perfectly wonderful regular, planar hexagons.

A downside of this broad view is that one loses connection with some of basic facts (for instance, "the measures of angles of a hexagon sum to 720 degrees") that hold only for the most-elementary figures that one first encounters in Geometry, but as @Joseph mentions, "[O]ften the way mathematics grows is when an intuitive/familiar concept evolves to get insight into a broader collection of ideas".

What I learned from Branko Grünbaum is that

Once you think you completely understand something (like the notion of "polygon"), a generalization will turn up that shows you that you're just getting started. This principle applies even when you've already taken this principle into account, which is to say: You're never done.

It's not a quote, but I think it summarizes his philosophy pretty well.

So, even if this stuff is well beyond the scope of your son's current coursework, it can be helpful to have some awareness that there's much more to mathematics than what's presented in any one textbook ... if only to encourage students who get bored in class to find ways of tweaking the concepts in ways that make them interesting. ("But what if some of my hexagon's vertices lifted out of the page?") This really is how mathematics often advances.

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Wow, I'm glad this wasn't just added as a comment. That definition of "hexagon" and the subsequent development really blew my mind. :) –  Jordan Gray Oct 8 '12 at 9:16
@Blue While I am big fan of flexible terminology I think it can also do "more harm" than good to use terms in a way that distorts their core or historical meaning. Thus, there is this nice paper about generalizing terms and ideas originally used for "straight" things to curves. Algorithmica, Volume 5, (1990), 421-457, Computational geometry in a curved world by D. Dobkin and Diane Souvaine. I think this is better done using adjectives in front of the core term rather than acting as if the original term covers everything. Thus, pseudolines generalize lines, etc. –  Joseph Malkevitch Oct 8 '12 at 18:03
@JosephMalkevitch: I'm not sure I've addressed your concern, but (I hope) I've clarified that I don't intend the phrase "line segment" to generalize to curves; only the term "edge" ... which was abstracted to mean whatever-it-takes-to-connect-this-vertex-to-that-one by combinatorialists drawing graphs on paper long before I arrived. Was there something else about the terminology in my answer that bothered you? (It wasn't my purpose to be super-formal; the Grünbaum paper you reference handles that nicely.) –  Blue Oct 8 '12 at 20:00
"if only to encourage students who get bored in class to find ways of tweaking the concepts in ways that make them interesting" - Hexaflexagons anyone? :) youtube.com/watch?v=VIVIegSt81k –  Danra Dec 11 '12 at 20:44

The diagram you drew is considered a hexagon but often the way mathematics grows is when an intuitive/familiar concept evolves to get insight into a broader collection of ideas. This has been especially true for geometry. The critical feature for a polygon is that straight line segments join up the the points involved. Early mentions of polygons implicitly assumed that one was dealing with convex polygons, one had distinct points, one had no three of the points on a straight line or no three consecutive points on a straight line, that different points could not sit on top of each other, and that the points all were located in a single plane. Now geometers often investigate situations where some of these conditions are loosened and as a result many exciting new geometrical phenomena have come to light.

For a fascinating look at this type of issue for polyhedra instead of polygons look at the wonderful paper of Branko Grünbaum, "Are your polyhedra the same as my polyhedra?"

http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf

It deals with the many different ways that the concept of a polyhedron can be defined.

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Check your kid's geometry syllabus: it may be he's being taught simple, basic geometry and thus n-gons are usually considered to be convex, i.e.: the interior (and perimeter, too) of the n-gon must contain the whole line segment connecting any two points in the interior. Under this agreement what you drew wouldn't qualify as hexagon or, perhaps more accurately, convex hexagon.

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Yes, It is Considered as a Hexagon. There is a difference between an Irregular Hexagon and a Regular Hexagon. A regular hexagon has sides that are segments of straight lines that are all equal in length. The interior angles are all equal with 120 degrees. An irregular hexagon has sides that may be of different lengths. It also follows that the interior angles are not all equal. Some interior angles may be greater than 180 degrees, but the sum of all interior angles is 720 degrees. Hope this gives u an idea about it.

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A closed polygon with 6 sides. Total Interior angles = (n-2) x 180 where n is the number of sides. One exterior angle = 360/n

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This doesn't seem to answer the original question. –  Rick Decker Oct 14 '12 at 20:00

For polygon the closing sides from start to end point after the desired number reached is said to be a polygon. Specific term come into action only when the condition like angle and length are taken into account....

say for example, i am dreaming to draw 6 sides closed (still its too a polygon with 6 sides)

I am drawing a circle, its total angle is 360.

Now i am planning to divide its total angle into 6 parts(sectors) I have made the imaginary lines for the angles inscribed. 360/6 = 60 degree

now i am joining the lines starting from center of circle to the circumference of it until i'm touching it.

now i am marking the points in which formed on the circumference of the circle due to the line joining from the center of circle to the circumference due to the angle i marked (60 degree)

Now join the marked points each other remove the circle you will get a perfect HEXAGON

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This doesn't seem to be an answer to the original poster's question: "Why the 6 angle requirement?" –  Rick Decker Oct 8 '12 at 13:31

It has six sides and six non-intersecting lines connecting them. Its interior is connected. Therefore, it's a hexagon.

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The above shown image is of irregular hexagon with uneven angles and edges.
Below shown image is for regular hexagon with even edges and angles Image. so you can divide hexagons into 2 subcategories. 1. even and 2. uneven.
So, in even its by default all edges and angles are similar.
and under uneven all angles and edges can be same or not.
All other hexagons come under uneven hexagons.

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It is really unclear what you mean by "even" and "uneven" here. Are you randomly switching your vocabulary from "regular and "irregular" to "even" and "uneven"? –  rschwieb Oct 18 '12 at 15:10
here even denotes equality. –  vidur punj Oct 19 '12 at 8:43
...equality of what? Side length? Angles? Both? Why not just stick with regular-irregular? –  rschwieb Oct 19 '12 at 11:49
Both, even angles means all angle are equal, even sides means all sides are equal, and even hexagon means hexagon with even edges and angles. –  vidur punj Oct 28 '12 at 5:39