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WikiProject Mathematics (Rated Start-class, Mid-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
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 Field:  Geometry
One of the 500 most frequently viewed mathematics articles.

Suggestion for improvements[edit]

I don't have time at the moment to do this, but this page could use substantial revision. I may try to get to this.

It's really great that people have put a lot of time into this page and that there are several constructions here-- that's what led me here in the first place. But now that they're here, it'd be nice to systemize the presentation and make it more unified.

after all, all the constructions (compass and straightedge constructions, that is) begin in the same way: construct the midpoint and the perpendicular to the given radius. the alternative method and that using the Carlyle circles follow each other for even longer. All the constructions could have, for example, the same notation for the same objects and common steps described in common. There's at least three other constructions that should be included in detail -- Euclid's; a variation where (using the notation of the alternative method) one goes through step 4 of the alternative method, then takes a circle centered on D through point O; two of the five vertices lie on the intersection of this circle with the original and point B is another. Finally, a very nice construction that might or might not be due to John H. Conway and seems the simplest of all: using the notation of the alternative method, go through step 3; take point E to be the intersection of segment AB with the circle centered on midpoint C passing through O and B. Now take the circle centered on A passing through E-- presto! The intersection of this circle with the original gives two vertices.

A construction of penta, given an edge would also be nice to include.

Additional comments: the proof given is kinda beside the point. A purely synthetic proof would be more in the spirit of compass and straightedge, and would be simpler and much less intimidating to some people. All of these constructions set up a golden section and then plop it around; all that's needed is that the golden section appears in the penta as advertised. There are many simple variations -- Euclid's is great.

The so-called direct method really seems kinda, well, stupid. Certainly not at all of the same spirit as the others.

In any case, nice material, but now that it's here, can be taken to a more professional standard. (talk) 20:09, 21 August 2010 (UTC)

One problem is at least on my two computers the pdf version refuses to print. I get a notice "Died Code 1" (Corrected post) Nicodemus (talk) 20:55, 29 June 2016 (UTC)

the pentagon[edit]

Excuse me a minute, but isn't it a bit thick to put the geometrical object pentagon aside for a building bearing its shape? Most people in this world will not associate the word "Pentagon" with a building, even though most USA-citizens probably do.

The thing that is on pentagon (shape) should be moved here, and the things here to pentagon (building). Need opinions about this before I do it.

-- Jörgen Nixdorf

Agreed. I thought the same as soon as I saw it. I was going to rename the articles pentagon and The Pentagon. In fact, it seems that articles with those names exist, so your work is half done. -- Heron

There I did it. Nixdorf 07:12 May 13, 2003 (UTC)

I fixed them. All of them. Nixdorf 10:44 May 14, 2003 (UTC)
Did you good:
Whew! Nicely done. -- Minesweeper

What is a "no salute, no cover" area? - Montréalais

"Cover" is military jargon for "hat". Normally a US serviceman must salute his superior officers whenever he (the subordinate) is wearing a cover, or is indoors, and must not do so otherwise. Covers are not usually worn indoors. Outdoors, a "no salute, no cover" area is one where you are not required to wear military millinery, and therefore not required to salute your superiors. --Heron

Thanks. If I weren't 10 minutes late for school I'd try to think of a graceful way of putting this into the article. - Montréalais

Done, but not gracefully. --Heron


I think that there should be a disambiguation for the term "Pentagon". I searched for it, expecting to find results on the building, and was brought to the shape. As Jörgen Nixdorf said, many USA citizens associate this word with the building, and so expect results on the building. But, of course the shape is also an important article. Typing "the pentagon" instead of "pentagon" is something that not a lot of people would consider doing. In short, I think this term needs an disambiguation with links to The Pentagon and Pentagon. I would do it myself, but I don't know how.

Actually, I think if I were going to look for info on the building, I would probably search for "the pentagon", and expect "pentagon" to lead to the shape. Even if not, only one extra click is needed to reach The Pentagon. If there were a disambig, you would need an extra click to arrive at either page, and the article about the shape would need an awkward name like Pentagon (shape). Also bear in mind that the building is named after the shape (I assume). Bistromathic 12:26, 20 August 2007 (UTC)
There is already a Pentagon (disambiguation) page. This article links to it in the intro, and also to The Pentagon for the US government building. It all looks fine to me . -- Steelpillow 21:11, 21 August 2007 (UTC)

Origami pentagon[edit]

I thought you might find this interesting: Shinobu (talk) 12:55, 29 September 2008 (UTC)

a pentagon has 5 side —Preceding unsigned comment added by (talk) 22:19, 13 January 2009 (UTC)

I think this is what you mena... Arydberg (talk) 15:36, 9 May 2016 (UTC)

Compass as measuring tool[edit]

Sorry I don't have time to do more than just mention this:

The "alternative method" described in the article is invalid as a classical contruction method for a pentagon, because it uses a compass as a measuring device.

This is a horribly common method shown for this. In fact the animated gif in this article is one of the very few methods I've seen that actually does it correctly. No wonder our kids are so messed up. :)Kid Bugs (talk) 17:42, 8 February 2009 (UTC)

Could you explain how, please? I'm one of the messed up youths of today, and can't seem to understand which step uses the compass as a measurer! Thanks a bunch! --Oskjoh (talk) 19:11, 1 November 2010 (UTC)

There is no verbal description for the main construction that is illustrated by the GIF. The GIF itself goes by too fast for it to be useful without any verbal description. The first "alternate construction" is a good one, but it has no GIF. The combination is confusing; throughout several visits to the page, I thought the first verbal description (i.e. the first "alternative") was attempting to describe the construction illustrated in the GIF. But the methods are fundamentally different, and I was suffering severe cognitive dissonance watching one construction and reading the description of another.

Could someone write a new description to go with the GIF? (talk) 20:57, 15 December 2009 (UTC)tad brennan

it also had a hidden vertez in it —Preceding unsigned comment added by (talk) 18:15, 23 March 2010 (UTC)

I have finished the "verbal description" as you pleased. (Richmond's Method) I do not understand the "hidden vertez" statement. Which method has the hidden vertex? Bon 062 (talk) 11:25, 29 August 2011 (UTC)

Statement of tiling result[edit]

I think the passage "A pentagon cannot appear in any tiling made by regular polygons. To prove a pentagon cannot form a regular tiling, 360 / 108 = 3 1/3, which is not a whole number. More difficult is proving a pentagon cannot be in any tiling made by regular polygons:" needs some work. (1) It's self-contradictory, saying that something is more difficult than itself. (2) The proof, "360 / 108 = 3 1/3, which is not a whole number", leaves out too much detail: why divide by 108 -- and what theorem is being appealed to here? Duoduoduo (talk) 20:06, 24 May 2010 (UTC)

The measure of a pentagon's angle is 108 degrees. Georgia guy (talk) 20:32, 24 May 2010 (UTC)
Right. The statement is self-contradictory. I'll be working on it. Bon 062 (talk) 05:32, 30 August 2011 (UTC)
A complete rotation measures 360 degrees. If 108-degree angles are placed together in an attempt to make a tessellation, only three angles measuring 324 degrees may fit. Therefore, leaving a gap, opposing the definition of tiling. (Tessellation) Bon 062 (talk) 09:47, 30 August 2011 (UTC)
The statement is not self-contradictory. The first statement, namely:

A pentagon cannot appear in any tiling made by regular polygons. To prove a pentagon cannot form a regular tiling, 360 / 108 = 3 1/3, which is not a whole number.

merely says that a regular tiling (a tiling with all congruent regular polygons) cannot have pentagons. There are only 3 such tilings; the triangular, square, and hexagonal tilings.

However, the statement:

More difficult is proving a pentagon cannot be in any tiling made by regular polygons

is about proving something stronger. This is that there are absolutely no tilings of regular polygons that contain at least one pentagon anywhere. There are only 5 polygons that appear at least once in any tiling of regular polygons; the triangle, square, hexagon, octagon, and dodecagon. Georgia guy (talk) 12:50, 30 August 2011 (UTC)

I can't find a proof online but it's easy OR. A regular pentagon must share its vertex with either a square and an isosogon or another pentagon and a dodecagon (stated here; other candidates are easy but tedious to eliminate). Each option forces a ring of ten pentagons round the larger shape, forcing ten more large shapes on the other side of each pair of pentagons. In either case, the ten outer large shapes overlap each other, proving that a pentagon cannot participate in a tiling of regular polygons. Certes (talk) 15:47, 11 December 2011 (UTC)
Two pentagons and a dodecagon don't produce 360 degrees. They produce 108 + 108 + 150 = 366 degrees; we need exactly 360 degrees. Now, assuming isosogon is a typo for icosagon (20 sides,) it's right that they add up to 360 degrees, but we can't make a tiling out of this vertex, as can be proven as follows:

The pentagon has an odd number of sides. Therefore, we cannot alternate the polygons that touch the sides of the pentagon perfectly between a square and an icosagon. It would require an even number of sides on the pentagon. Is this right?? Georgia guy (talk) 15:51, 11 December 2011 (UTC)

Thank you for improving the proof, that's much clearer. Sorry, I meant decagon not dodecagon above. The limited options for a pentagon (4.5.20 and 5.5.10) do require alternate neighbours to be 4,20,4... or 5,10,5... which gives the proof we need. (Pedantically, I think you can have other odd polygons which don't alternate. Imagine a trihexagonal tiling with every third hexagon replaced by six new triangles; each original triangle is then surrounded by two hexagons and a new triangle. But none of that breaks this proof.) Certes (talk) 18:00, 11 December 2011 (UTC)
(edited to remove my ramble which is irrelevant, because it was shown that each vertex has only three polygons. Certes (talk) 19:05, 11 December 2011 (UTC)

Yet another construction[edit]

I moved the following construction method from the article page, mainly because it was in the wrong section, but also because we already have many construction methods, and this one needs some editing to be in acceptable format. Do we already have more than enough construction methods, or is there value in adding this one?

"Pentagon also can be drawn using general method available with Engineering Drawing Practice. The method involves the following procedure.

  • 1.Take side length as t (Let us take t-50mm)
  • 2. Draw side length t and call that line as AB
  • 3. At B, generate a perpendicular BC such that AB=BC.
  • 4. Join C and A.
  • 5. Bisect the line AB and generate the line. This line touches line joining C and A at one point. Call it as Point 4.
  • 6. Join C and A with an arc with B as Center and AB as radius. This arc cuts the bisecting line at one point. Call it as Point 6.
  • 7. Now bisect point 4 and point 6 line to get point 5.
  • 8. With Point 5 as center, 5A as radius draw a Circle with Point 5 as center.
  • 9. Now start getting Points, by cutting along the circle with B as center get points x, y z with the same approach.
  • join ABXYZ to get a pentagon."

Dbfirs 07:48, 3 November 2010 (UTC)

Addition: radius of circumscribed circle (circumradius) and radius of inscribed circle (apothem)[edit]

I think these 2 values (in relation with the lenght of the regular pentagon's side) should be presented here one after the another like they are presented in the equilateral triangle article. I made the same request at the square article. Bigshotnews 06:02, 16 February 2011 (UTC) — Preceding unsigned comment added by Bigshotnews (talkcontribs)

Draw Pentagon and 5 Point Star Simplest method[edit]

To draw a Pentagon and a 5 pointed star using only a ruler and pair of compasses. [1] Decide the diameter (d) of the pentagon and star that you require, then with compasses draw a circle of that diameter of radius R. (R =d/2). [2] Multiply R x 0.309, call the result r. [3] Draw a circle radius r using the same center as the first circle. [4] Pick any point on the circumference of the first circle (A) and draw a line forming a tangent to the smaller circle. Mark (B) as the point where the extension of this tangent passes through the large circle. [5] Now draw another tangent to the smaller circle from point (B), mark the intersection of its extension with the larger circle as point (C). [6] Now draw another tangent to the smaller circle from point (C), mark the intersection of its extension with the larger circle as point (D). [7] Now draw another tangent to the smaller circle from point (D), mark the intersection of its extension with the larger circle as point (E). [8] The extension of a tangent from point (E) to the smaller circle will end at your original point (A). [9] To create a pentagon from the 5 pointed star draw lines linking ABCDE. Ron Neale62.56.51.38 (talk) 11:02, 5 March 2011 (UTC)

... but most mathematical constructions are restricted to a straight edge and compasses. There are even easier methods if you are allowed to make measurements (and yours are only approximate anyway). Dbfirs 22:09, 6 March 2011 (UTC)

I think this is the quickest method if you need to accurately define the size of the your pentagon/star in advance. I can provide the number 0.309 to as many decimal places as you would like. Then as with all other methods iy becomes a matter of how accurately you can draw. To aid accuracy make sure your "tangents" pass through the lines that describe your circle, i.e. they are point tangents- Ron62.56.48.211 (talk) 10:20, 25 March 2011 (UTC)

No, 0.309 has only three decimal places. It it not the number that you were intending to mention, which I also can calculate to any number of decimal places, but I cannot give its exact value without using square roots. The problem is that my ruler is accurate to less than one part in a thousand, so this method of construction cannot be accurate using my ruler, The advantage of constructing rather than measuring your number is that the result does not depend on the accuracy of markings on your measuring instrument. Your method is interesting, and will be worth including if you can find a way to construct the smaller circle rather than measuring it. Your method as stated is not the simplest because you could just measure each side of calculated length within your large circle. Dbfirs 06:30, 30 August 2011 (UTC)

Deciding whether a pentagon is cyclic[edit]

  • A triangle ABC is always cyclic.
  • A quadrilateral ABCD is cyclic if and only if A+C = 180 degrees (B+D will also always be 180 degrees.)
  • A pentagon ABCDE is cyclic if and only if...

Georgia guy (talk) 17:49, 29 March 2012 (UTC)

Interesting question. I'm re-posting it at Wikipedia:Reference desk/Mathematics#March 29. Duoduoduo (talk) 19:28, 29 March 2012 (UTC)
See Wikipedia:Reference_desk/Archives/Mathematics/2012_March_30#Deciding_whether_a_pentagon_is_cyclic. Double sharp (talk) 14:22, 15 April 2012 (UTC)


I've removed the Ref.Improve tag from four and a half years ago because there are ten in-line citations and references. More references could certainly be added, but Wikipedia has thousands of untagged articles that are unreferenced, and I don't think any of the information in this article is in dispute. Dbfirs 22:44, 29 January 2013 (UTC)

Regular convex pentagon?[edit]

Regular pentagon can be not convex? ru:МетаСкептик12 — Preceding unsigned comment added by МетаСкептик12 (talkcontribs) 09:48, 24 February 2013 (UTC)

Regular implies that all angles are equal, so only a convex configuration is possible in conventional Euclidean geometry. However a pentagram is sometimes considered to be a star pentagon, and is also regular. Dbfirs 15:39, 24 February 2013 (UTC)
When a pentagram is considered to be a polygon, it's always considered a star pentagon, and so a regular pentagram with 36° angles and all sides equal would indeed be a regular pentagon. But normally we reserve "-gon" for the convex forms {p/1} and use "-gram" only for the nonconvex forms {p/q}. Double sharp (talk) 16:17, 24 February 2013 (UTC)

Please clear this up[edit]

In a Robbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all the diagonals must be rational.[10]

Well, if all diagonals are rational, then all diagonals can be made irrational by changing the unit. For example, if all diagonals are a rational number of inches, then all diagonals are an irrational number of units equal to an inch times the square root of 2. Georgia guy (talk) 13:59, 1 May 2014 (UTC)

Yes, but the area and side lengths must also be rational. Double sharp (talk) 14:52, 14 March 2015 (UTC)

Not all pictures in the gallery are pentagons[edit]

Some of these objects are star pentagons, so they technically qualify for this article but would be more appropriate to the pentagram or star pentagon articles. Others, like the apple gynoecium, are merely objects with fivefold symmetry and not pentagons at all. --Heron (talk) 13:54, 17 August 2014 (UTC)

Other kinds of pentagons[edit]

Shouldn't this article also include references to other kinds of pentagons, i.e., equilateral pentagons, which are commonly encountered? As well as any other kinds of significant pentagons? 8bitW (talk) 19:51, 28 January 2016 (UTC)

Yes, there should at least be a link to that article (I've added it under "see also", but you could add a section if you think it's important enough). Which other significant pentagons did you have in mind? Dbfirs 20:04, 28 January 2016 (UTC)
Actually, I'm not aware of any others, but I figured I'd leave it out there in case there are any others that I'm not aware of. :-D I came to this article specifically looking information on the equilateral pentagon (although I didn't know its name), so I was a bit frustrated having to do some sleuthing to find what I was looking for. It might be wise to put a small line in the lede referencing the rather-common equilateral pentagon, briefly explaining the differences between the two, since it is in fact so commonly known.
On a side note, the regular polygon page could use some disambiguation as well. It seems that most of these geometry-related articles are written at somewhat of an expert level. I have a college degree, made it all the way through Calc 1 in high school, etc., but I can hardly make heads or tails out of a lot of these articles. At the very least, the ledes should provide an adequate overview in layman's terms, so that even those who choose not to peruse the details can still feel they have a decent grasp of the topic. 8bitW (talk) 23:23, 28 January 2016 (UTC)
I added a small line to the lede disambiguating the topic from an equilateral pentagon, but now that I think about it, isn't the entire topic of the article skewed? Since a pentagon is any 5-sided polygon, there are as many kinds of pentagons as there are polygons. The article, and especially the lede, however, is overwhelmingly skewed towards regular pentagons. I'm afraid the complexity of this topic is beyond my ability to resolve. Any geometry experts out there? 8bitW (talk) 23:30, 28 January 2016 (UTC)

Equilateral pentagon picture[edit]

Is it possible to change the alignment on the illustration for the section "Equilateral pentagon"? It's a bit crowded now. (Sorry, I'm not much of an expert with the wiki software / code.) 8bitW (talk) 18:35, 29 January 2016 (UTC)

Pentagons as Forts[edit]

Many forts are shaped as pentagons. The reason is that the parapets built out at the corners allow a better field of fire directed at the base of the walls. Arydberg (talk) 12:29, 15 April 2016 (UTC)

Method of Demetrius[edit]

Area of a regular pentagon.

You should try to get away from trig functions. Here is another method of finding the area of a pentagon. This proof uses the fact that Phi ( the golden ratio) occurs 3 times in relating the ratio of different sides in a pentagon with an inscribed 5 point star.

Method of Demetrius Let AT equal the area of the large pentagon. Let the sides of the small center pentagon equal 1 and let it's area equal a . Then the two equal sides of the point triangles have a length of Phi. The sides of the large pentagon are equal to Phi^2.

Take the area of any triangular point on the inscribed 5 point star and add it to the area of the triangle to it's immediate left. This sum is an isosceles triangle which has the two equal sides equal to Phi^2 and a base of Phi. Let the area of this triangle equal T.

so AT = 5 x T + a

but also AT = a x Phi^4 ( ratio of sides is Phi^2 area is proportional to square of side ratio )

or a x Phi^4 = 5 x T + a

                  a x Phi^4 - a = 5 x T 
                  a( Phi^4 - 1 )  =  5 x T 
                  a = 5 x T /(Phi^4  - 1 )  

T is given by (base/4) x sqrt ( 4 x side^4 - base^2) if you wish to go further. — Preceding unsigned comment added by Arydberg (talkcontribs) 19:03, 19 April 2016 (UTC) Arydberg (talk) 10:35, 4 May 2016 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Pentagon/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

General expansion needed. Geometry guy 21:40, 17 June 2007 (UTC)

I love Science and mathamatics but there is one thing i do not Know was is the Formula of A

Starjester1009:38, Starjester10 (talk) 09:43, 26 January 2009 (UTC)

Last edited at 09:43, 26 January 2009 (UTC). Substituted at 02:28, 5 May 2016 (UTC)


Can a pentagon also be called a quintgon? (talk) 13:09, 8 April 2017 (UTC)

No. Please stop doing using Wikipedia as a discussion forum for original research. You can read more about original research, and why it is not allowed in Wikipedia, at the page WP:OR. If you'd like to help contribute positively to Wikipedia, there is a good capsule guide to Wikipedia's policies at WP:PILLARS. -- The Anome (talk) 15:11, 8 April 2017 (UTC)
@ See WP:NOTNEO. --Thnidu (talk) 15:39, 8 April 2017 (UTC)


I noticed that there could be some confusion between the pentagon (shape) and the intelligence base going by the same name. It's probably too much hassle but thanks for taking the time to read this! — Preceding unsigned comment added by Ooh Saad (talkcontribs) 13:59, 28 May 2020 (UTC)