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{{Short description|Shape with five sides}}
{{About|the geometric figure|the headquarters of the United States Department of Defense|The Pentagon|other uses}}
{{Infobox Polygon
| name          = Pentagon
| image         = 5-gon cyclic 01.svg
| caption       = A [[cyclic polygon|cyclic]] pentagon
| edges         = 5
| sides         =
| angle         = }}

In [[geometry]], a '''pentagon''' ({{ety|el|''πέντε'' (pente)|five||''γωνία'' (gonia)|angle}}<ref>"pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.</ref>) is any five-sided [[polygon]] or 5-gon.  The sum of the [[internal angle]]s in a [[simple polygon|simple]] pentagon is 540°.

A pentagon may be simple or [[list of self-intersecting polygons|self-intersecting]]. A self-intersecting ''regular pentagon'' (or ''[[star polygon|star]] pentagon'') is called a [[pentagram]].

== Regular pentagons ==
{{Regular polygon db|Regular polygon stat table|p5}}
[[File:Regular pentagon 1.svg|thumb|Side (<math>t</math>), [[circumradius]] (<math>R</math>), inscribed circle radius (<math>r</math>), height {{Nowrap|(<math>R + r</math>)}}, width/diagonal {{Nowrap|(<math>\varphi t</math>)}}]]

A ''[[regular polygon|regular]] pentagon'' has [[Schläfli symbol]] {5} and [[interior angle]]s of 108°.

A ''[[regular polygon|regular]] pentagon'' has five lines of [[reflectional symmetry]], and [[rotational symmetry]] of order 5 (through 72°, 144°, 216° and 288°). The [[diagonal]]s of a [[convex polygon|convex]] regular pentagon are in the [[golden ratio]] to its sides. Given its side length <math>t,</math> its height <math>H</math> (distance from one side to the opposite vertex), width <math>W</math> (distance between two farthest separated points, which equals the diagonal length <math>D</math>) and circumradius <math>R</math> are given by:

:<math>\begin{align}
    H &= \frac{\sqrt{5+2\sqrt{5}}}{2}~t \approx 1.539~t, \\
    W=
    D &= \frac{1 + \sqrt5}{2}~t\approx 1.618~t, \\
    W &= \sqrt{2-\frac{2}{\sqrt{5}}} \cdot H\approx 1.051~H, \\
    R &= \sqrt{\frac{5+\sqrt{5}}{10}} t\approx 0.8507~t, \\
    D &= R\ {\sqrt { \frac {5 + \sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.902~R.
\end{align}</math>

The area of a convex regular pentagon with side length <math>t</math> is given by
:<math>\begin{align}
   A &= \frac{{t^2 \sqrt {25 + 10\sqrt 5} }}{4} = \frac{5t^2 \tan 54^\circ}{4} \\
     &= \frac{\sqrt{5(5 + 2\sqrt{5})} \;t^2}{4} = \frac{t^2 \sqrt{4 \varphi^5 + 3}}{4}
 \approx 1.720~t^2 
\end{align}</math>

If the circumradius <math>R</math> of a regular pentagon is given, its edge length <math>t</math> is found by the expression
:<math>t = R\ {\sqrt { \frac {5 - \sqrt{5}}{2}} } = 2R\sin 36^\circ = 2R\sin\frac{\pi}{5} \approx 1.176~R,</math>

and its area is

:<math>A = \frac{5R^2}{4}\sqrt{\frac{5 + \sqrt{5}}{2}};</math>

since the area of the circumscribed circle is <math>\pi R^2,</math> the regular pentagon fills approximately 0.7568 of its circumscribed circle.

=== Derivation of the area formula ===
The area of any regular polygon is:
:<math>A = \frac{1}{2}Pr</math>

where ''P'' is the perimeter of the polygon, and ''r'' is the [[inradius]] (equivalently the [[apothem]]). Substituting the regular pentagon's values for ''P'' and ''r'' gives the formula
:<math>A = \frac{1}{2} \cdot 5t \cdot \frac{t\tan\mathord\left(\frac{3 \pi}{10}\right)}{2} = \frac{5t^2\tan\mathord\left(\frac{3 \pi}{10}\right)}{4}</math>
with side length ''t''.

=== Inradius ===

Similar to every regular convex polygon, the regular convex pentagon has an [[tangential polygon|inscribed circle]]. The [[apothem]], which is the radius ''r'' of the inscribed circle, of a regular pentagon is related to the side length ''t'' by

:<math>r = \frac{t}{2\tan\mathord\left(\frac{\pi}{5}\right)} = \frac{t}{2\sqrt{5 - \sqrt{20}}} \approx 0.6882 \cdot t.</math>

=== Chords from the circumscribed circle to the vertices ===
Like every regular convex polygon, the regular convex pentagon has a [[circumscribed circle]]. For a regular pentagon with successive vertices A, B, C, D, E,  if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.

=== Point in plane ===
For an arbitrary point in the plane of a regular pentagon with circumradius <math>R</math>, whose distances to the centroid of the regular pentagon and its five vertices are <math>L</math> and <math>d_i</math>
respectively, we have<ref name="Mamuka">{{cite journal |last1=Meskhishvili |first1=Mamuka |date=2020 |title=Cyclic Averages of Regular Polygons and Platonic Solids |url=https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065 |journal=Communications in Mathematics and Applications |volume=11 |pages=335–355 |doi=10.26713/cma.v11i3.1420 |doi-broken-date=14 December 2024 |arxiv=2010.12340 }}</ref>

:<math>\begin{align}
  \textstyle \sum_{i=1}^5 d_i^2 &= 5\left(R^2 + L^2\right), \\
  \textstyle \sum_{i=1}^5 d_i^4 &= 5\left(\left(R^2 + L^2\right)^2 + 2R^2 L^2\right), \\
  \textstyle \sum_{i=1}^5 d_i^6 &= 5\left(\left(R^2 + L^2\right)^3 + 6R^2 L^2 \left(R^2 + L^2\right)\right), \\
  \textstyle \sum_{i=1}^5 d_i^8 &= 5\left(\left(R^2 + L^2\right)^4 + 12R^2 L^2 \left(R^2 + L^2\right)^2 + 6R^4 L^4\right).
\end{align}</math>

If <math>d_i</math> are the distances from the vertices of a regular pentagon to any point on its circumcircle, then<ref name= Mamuka />

:<math>3\left(\textstyle \sum_{i=1}^5 d_i^2\right)^2 = 10 \textstyle \sum_{i=1}^5 d_i^4 .</math>

=== Geometrical constructions ===

The regular pentagon is constructible with [[compass and straightedge]], as 5 is a [[Fermat prime]]. A variety of methods are known for constructing a regular pentagon. Some are discussed below.

==== Richmond's method ====
[[File:Richmond pentagon 1.PNG|thumb]]
One method to construct a regular pentagon in a given circle is described by Richmond<ref name=Richmond>{{Cite journal|last=Richmond|first= Herbert W. |title=A Construction for a Regular Polygon of Seventeen Sides | journal=The Quarterly Journal of Pure and Applied Mathematics|volume= 26|pages= 206–207|year= 1893 | url=https://books.google.com/books?id=bcMKAAAAIAAJ&pg=PA206 }}</ref> and further discussed in Cromwell's ''[[Polyhedra (book)|Polyhedra]]''.<ref>{{cite book |title=Polyhedra |title-link=Polyhedra (book)|author=Peter R. Cromwell |date=22 July 1999|publisher=Cambridge University Press |at=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA63 p. 63] |isbn=0-521-66405-5}}</ref>

The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point ''C'' and a midpoint ''M'' is marked halfway along its radius. This point is joined to the periphery vertically above the center at point ''D''. Angle ''CMD'' is bisected, and the bisector intersects the vertical axis at point ''Q''. A horizontal line through ''Q'' intersects the circle at point ''P'', and chord ''PD'' is the required side of the inscribed pentagon.

To determine the length of this side, the two right triangles ''DCM'' and ''QCM'' are depicted below the circle. Using [[Pythagoras' theorem]] and two sides, the hypotenuse of the larger triangle is found as <math>\scriptstyle \sqrt{5}/2</math>. Side ''h'' of the smaller triangle then is found using the [[half-angle formula]]:
:<math>\tan(\phi/2) = \frac{1 - \cos(\phi)}{\sin(\phi)} \ ,</math>

where cosine and sine of ''ϕ'' are known from the larger triangle. The result is:
:<math>h = \frac{\sqrt 5 - 1}{4} \ .</math>

If DP is truly the side of a regular pentagon, <math>m \angle\mathrm{CDP} = 54^\circ</math>, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos<sup>2</sup>(54°), and CQ = 1 − 2cos<sup>2</sup>(54°), which equals −cos(108°) by the cosine [[double angle formula]].  This is the cosine of 72°, [[exact trigonometric values#Common angles|which equals]] <math>\left(\sqrt 5 - 1\right)/4</math> as desired.

==== Carlyle circles ====
{{Main|Carlyle circle}}
[[File:Regular Pentagon Using Carlyle Circle.gif|thumb|Method using Carlyle circles]]
The Carlyle circle was invented as a geometric method to find the roots of a [[quadratic equation]].<ref name=Weisstein>{{cite book |title=CRC concise encyclopedia of mathematics |author=Eric W. Weisstein |page=329 |url=https://books.google.com/books?id=Zg1_QZsylysC&pg=PA329 |isbn=1-58488-347-2 |year=2003 |edition =2nd |publisher=CRC Press}}</ref> This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:<ref name=DeTemple>{{cite journal|last=DeTemple|first=Duane W.|title=Carlyle circles and Lemoine simplicity of polygon constructions|journal=The American Mathematical Monthly|date=Feb 1991|volume=98|issue=2|pages=97–108|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf|archiveurl=https://web.archive.org/web/20151221113614/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#3|doi=10.2307/2323939|jstor=2323939|archive-date=2015-12-21 |url-status=dead}}</ref>

# Draw a [[circle]] in which to inscribe the pentagon and mark the center point ''O''.
# Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point ''B''.
# Construct a vertical line through the center. Mark one intersection with the circle as point ''A''.
# Construct the point ''M'' as the midpoint of ''O'' and ''B''.
# Draw a circle centered at ''M'' through the point ''A''. Mark its intersection with the horizontal line (inside the original circle) as the point ''W'' and its intersection outside the circle as the point ''V''.
# Draw a circle of radius ''OA'' and center ''W''. It intersects the original circle at two of the vertices of the pentagon.
# Draw a circle of radius ''OA'' and center ''V''. It intersects the original circle at two of the vertices of the pentagon.
# The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
Steps 6–8 are equivalent to the following version, shown in the animation:

: 6a. Construct point F as the midpoint of O and W.

: 7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is the rightmost intersection of the horizontal line with the original circle.

: 8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.

==== Euclid's method ====
[[File:01-Pentagon-Euklid Animation.gif|thumb|Euclid's method for pentagon at a given circle, using the [[Golden triangle (mathematics)|golden triangle]], animation 1 min 39 s]]
A regular pentagon is [[constructible polygon|constructible]] using a [[compass and straightedge]], either by inscribing one in a given circle or constructing one on a given edge. This process was described by [[Euclid]] in his ''[[Euclid's Elements|Elements]]'' circa 300 BC.<ref name=Martin>{{cite book |title=Geometric constructions |author=George Edward Martin |url=https://books.google.com/books?id=ABLtD3IE_RQC&pg=PA6 |page=6 |isbn=0-387-98276-0 |year=1998 |publisher=Springer}}</ref><ref name=Fitzpatrick>{{cite book |title=Euklid's Elements of Geometry, Book 4, Proposition 11 |translator=Richard Fitzpatrick | url=https://farside.ph.utexas.edu/books/Euclid/Elements.pdf#page=120&zoom=auto,-205,774|isbn=978-0-615-17984-1| page=119 |year=2008|last1=Fitzpatrick |first1=Richard |publisher=Lulu.com }}</ref>

=== Physical construction methods ===
[[File:Overhand-folded-ribbon-pentagon.svg|thumb|Overhand knot of a paper strip]]
* A regular pentagon may be created from just a strip of paper by tying an [[overhand knot]] into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a [[pentagram]] when backlit.<ref>''[[Mathematical Models (Cundy and Rollett)|Mathematical Models]]'' by [[Martyn Cundy|H. Martyn Cundy]] and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 57.</ref>
* Construct a regular [[hexagon]] on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to make a [[pentagonal pyramid]]. The base of the pyramid is a regular pentagon.

=== Symmetry ===
[[File:Symmetries of pentagon.png|thumb|Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.]]

The ''regular pentagon'' has [[dihedral symmetry|Dih<sub>5</sub> symmetry]], order 10. Since 5 is a [[prime number]] there is one subgroup with dihedral symmetry: Dih<sub>1</sub>, and 2 [[cyclic group]] symmetries: Z<sub>5</sub>, and Z<sub>1</sub>.

These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref> Full symmetry of the regular form is '''r10''' and no symmetry is labeled '''a1'''. The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars), and '''i''' when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g5''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.

=== Regular pentagram ===
{{Main|Pentagram}}
A pentagram or pentangle is a [[regular polygon|regular]] [[star polygon|star]] pentagon. Its [[Schläfli symbol]] is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the [[Pentagram#Golden ratio|sides of the two pentagons]] are in the [[golden ratio]].

== Equilateral pentagons ==
{{Main|Equilateral pentagon}}
[[File:Equilateral pentagon.SVG|thumb|class=skin-invert-image|Equilateral pentagon built with four equal circles disposed in a chain.]]

An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique [[up to]] similarity, because it is equilateral and it is equiangular (its five angles are equal).

== Cyclic pentagons ==

A [[cyclic polygon|cyclic]] pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon.  The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a [[septic equation]] whose coefficients are functions of the sides of the pentagon.<ref>Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/CyclicPentagon.html]</ref><ref>{{cite journal |last=Robbins |first=D. P. |date=1994 |title=Areas of Polygons Inscribed in a Circle |journal=[[Discrete and Computational Geometry]] |volume=12 |issue=2 |pages=223–236 |doi=10.1007/bf02574377|doi-access=free }}</ref><ref>{{cite journal |last=Robbins |first=D. P. |date=1995 |title=Areas of Polygons Inscribed in a Circle |journal=The American Mathematical Monthly |volume=102 |issue=6 |pages=523–530 |doi=10.2307/2974766|jstor=2974766 }}</ref>

There exist cyclic pentagons with rational sides and rational area; these are called [[Robbins pentagon]]s. It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational.<ref>*{{citation
 | last1 = Buchholz
 | first1 = Ralph H.
 | last2 = MacDougall
 | first2 = James A.
 | doi = 10.1016/j.jnt.2007.05.005
 | issue = 1
 | journal = [[Journal of Number Theory]]
 | mr = 2382768
 | pages = 17–48
 | title = Cyclic polygons with rational sides and area
 | volume = 128
 | year = 2008
 | doi-access = free
 }}.</ref>

== General convex pentagons ==

For all convex pentagons with sides <math>a, b, c, d, e</math> and diagonals <math>d_1, d_2, d_3, d_4, d_5</math>, the following inequality holds:<ref name="Crux">''Inequalities proposed in “[[Crux Mathematicorum]]”'', [http://www.imomath.com/othercomp/Journ/ineq.pdf].</ref>{{rp|p.75,#1854}}

:<math>3(a^2+b^2+c^2+d^2+e^2) > d_1^2+d_2^2+d_3^2+d_4^2+d_5^2</math>.

== Pentagons in tiling ==
{{Main|Pentagon tiling}}
[[File:2-d pentagon packing dual.svg|thumb|right|The [[packing constant|best-known packing]] of equal-sized regular pentagons on a plane is a [[double lattice]] structure which covers 92.131% of the plane.]]
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a [[regular tiling]] (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that {{nowrap|360° / 108° {{=}} 3{{frac|1|3}}}} (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them.  More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:

The maximum known [[packing density]] of a regular pentagon is <math>(5-\sqrt 5)/3\approx 0.921</math>, achieved by the [[double lattice]] packing shown. In a preprint released in 2016, [[Thomas Callister Hales|Thomas Hales]] and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has the optimal density among all packings of regular pentagons in the plane.<ref>{{citation | last1 = Hales | first1 = Thomas | author1-link = Thomas Callister Hales | last2 = Kusner | first2 = Wöden | arxiv = 1602.07220 | date = September 2016 | title = Packings of regular pentagons in the plane}}</ref>

There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with&nbsp;3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around the pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all {{nowrap|(360 − 108) / 2 {{=}} 126°}}. To find the number of sides this polygon has, the result is {{nowrap|360 / (180 − 126) {{=}} 6{{frac|2|3}}}}, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.

There are 15 classes of pentagons that can [[Pentagonal tiling#Monohedral convex pentagonal tilings|monohedrally tile the plane]]. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry.
{| class=wikitable
|+ 15 monohedral pentagonal tiles
!1||2||3||4||5
|- align=center
|[[File:Prototile p5-type1.png|120px]]
|[[File:Prototile p5-type2.png|120px]]
||[[File:Prototile p5-type3.png|120px]]
||[[File:Prototile p5-type4.png|120px]]
||[[File:Prototile p5-type5.png|120px]]
|- align=center
!6||7||8||9||10
|- align=center
|[[File:Prototile p5-type6.png|120px]]
||[[File:Prototile p5-type7.png|120px]]
||[[File:Prototile p5-type8.png|120px]]
|[[File:Prototile p5-type9.png|120px]]
||[[File:Prototile p5-type10.png|120px]]
|- align=center
!11||12||13||14||15
|- align=center
|[[File:Prototile p5-type11.png|120px]]
||[[File:Prototile p5-type12.png|120px]]
||[[File:Prototile p5-type13.png|120px]]
|[[File:Prototile p5-type14.png|120px]]
|[[File:Prototile p5-type15.png|120px]]
|}

== Pentagons in polyhedra ==

{| class=wikitable width=840
![[Icosahedral symmetry|I<sub>h</sub>]]
![[Pyritohedral symmetry|T<sub>h</sub>]]
![[Tetrahedral symmetry|T<sub>d</sub>]]
![[Octahedral symmetry|O]]
![[Icosahedral symmetry|I]]
![[Dihedral symmetry in three dimensions|D<sub>5d</sub>]]
|-
|[[File:Dodecahedron.svg|120px]]
|[[File:Pyritohedron.png|120px]]
|[[File:Tetartoid.png|120px]]
|[[File:Pentagonalicositetrahedronccw.jpg|120px]]
|[[File:Pentagonalhexecontahedronccw.jpg|120px]]
|[[File:Pentagonal truncated trapezohedron.png|120px]]
|- align=center
|[[Dodecahedron]]
|[[Pyritohedron]]
|[[Tetartoid]]
|[[Pentagonal icositetrahedron]]
|[[Pentagonal hexecontahedron]]
|[[Truncated trapezohedron]]
|}

== Pentagons in nature ==

=== Plants ===
<gallery>
File:BhindiCutUp.jpg|Pentagonal cross-section of [[okra]].
File:Morning Glory Flower.jpg|[[Morning glories]], like many other flowers, have a pentagonal shape.
File:Rafflesia arnoldi - panoramio.jpg|[[wikt:perigone|Perigone]] tube of ''[[Rafflesia]]'' flower.
File:Sterappel dwarsdrsn.jpg|The [[gynoecium]] of an [[apple]] contains five carpels, arranged in a [[five-pointed star]]
File:Carambola Starfruit.jpg|[[carambola|Starfruit]] is another fruit with fivefold symmetry.
</gallery>

=== Animals ===
<gallery>
File:Oreaster reticulatus201905mx.jpg|A [[sea star]]. Many [[echinoderms]] have fivefold radial symmetry.
File:Sea Urchin Endoskeleton.jpg|Another example of echinoderm, a [[sea urchin]] endoskeleton.
File:Haeckel Ophiodea.jpg|An illustration of [[brittle stars]], also echinoderms with a pentagonal shape.
</gallery>

=== Minerals ===
<gallery>
File:Ho-Mg-ZnQuasicrystal.jpg|A Ho-Mg-Zn icosahedral [[quasicrystal]] formed as a pentagonal [[dodecahedron]]. The faces are true regular pentagons.
File:Pyrite elbe.jpg|A [[pyritohedron|pyritohedral]] crystal of [[pyrite]]. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
File:Au-Gold-2-1-16 Miass 2018-03-01 11-05-41 V3.gif|A [[Fiveling]] of gold, half a centimeter tall.
</gallery>

== Other examples ==
<gallery>
File:The Pentagon January 2008.jpg|[[The Pentagon]], headquarters of the [[United States Department of Defense]].
File:Home base of baseball field in Třebíč, Třebíč District.jpg|[[Home plate]] of a [[baseball field]]
</gallery>

== See also ==
* [[Associahedron]]; A pentagon is an order-4 associahedron
* [[Dodecahedron]], a polyhedron whose regular form is composed of 12 pentagonal faces
* [[Golden ratio]]
* [[List of geometric shapes]]
* [[Pentagonal number]]s
* [[Pentagram]]
* [[Pentagram map]]
* [[Pentastar]], the Chrysler logo
* [[Pythagoras' theorem#Similar figures on the three sides|Pythagoras' theorem]]
* [[Exact trigonometric values|Trigonometric constants for a pentagon]]

== In-line notes and references ==
{{Reflist|30em}}

== External links ==
{{Wiktionary|pentagon}}
{{Commons category}}
* {{MathWorld|title=Pentagon|urlname=Pentagon}}
* [http://www.mathopenref.com/constinpentagon.html Animated demonstration] constructing an inscribed pentagon with compass and straightedge.
* [https://web.archive.org/web/20080616063532/http://www.opentutorial.com/Construct_a_pentagon How to construct a regular pentagon] with only a compass and straightedge.
* [https://web.archive.org/web/20080701200220/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#knot How to fold a regular pentagon] using only a strip of paper
* [http://www.mathopenref.com/pentagon.html Definition and properties of the pentagon], with interactive animation
* [https://www.maa.org/book/export/html/116449 Renaissance artists' approximate constructions of regular pentagons] {{Webarchive|url=https://web.archive.org/web/20210413193914/https://www.maa.org/book/export/html/116449 |date=2021-04-13 }}
* [http://whistleralley.com/polyhedra/pentagon.htm Pentagon.] How to calculate various dimensions of regular pentagons.
<br />

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[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]
[[Category:5 (number)]]
[[Category:Elementary shapes]]