Pentagon
23930
225494988
2008-07-14T00:13:51Z
189.25.67.99
Give expression of the edge length of a regular pentagon incribed to a circle.
{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular pentagon
|-
|align=center colspan=2|[[Image:Pentagon.svg|250px]]<BR>A regular pentagon, {5}
|-
|bgcolor=#e7dcc3|[[Edge (geometry)|Edge]]s and [[Vertex (geometry)|vertices]]||5
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{5}
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]||[[Image:CDW_ring.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]]
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||[[Dihedral symmetry|Dihedral]] (D<sub>5</sub>)
|-
|bgcolor=#e7dcc3|[[Area]]<BR>(with ''t''=edge length)||<math>\frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4}</math><BR><math> \approx 1.720477401 t^2.</math>
|-
|bgcolor=#e7dcc3|[[Internal angle]]<BR>([[degree (angle)|degree]]s)||108°
|}
{{wiktionarypar|pentagon}}{{two other uses|the geometric figure|the headquarters of the United States Department of Defense|The Pentagon}}
In [[geometry]], a '''pentagon''' is any five-sided [[polygon]]. A pentagon may be simple or self-intersecting. The [[internal angle]]s in a simple pentagon total 540°.
== Regular pentagons ==
The term ''pentagon'' is commonly used to mean a '''regular convex pentagon''', where all sides are equal and all interior angles are equal (to 108°). Its [[Schläfli symbol]] is {5}.
The area of a regular convex pentagon with side length ''t'' is given by
<math>A = \frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4} = \frac{5t^2 \cdot \tan(54^\circ)}{4}\ \approx 1.720477401 t^2.</math>
A '''[[pentagram]]''' 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]].
When a regular pentagon is inscribed in a circle with radius <math>R</math>, its edge length <math>t</math> is given by the expression <math>t = R\ {\sqrt { \frac {5-\sqrt{5}}{2}} }\ \approx 1.17557050458 R</math>.
{{-}}
=== Construction ===
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.
One method to construct a regular pentagon in a given circle is as follows:
[[Image:Pentagon construct.gif|center|frame|Construction of a regular pentagon]]
An alternative method is this:
[[Image:pentagon-construction.svg|thumb|Constructing a pentagon]]
#Draw a [[circle]] in which to inscribe the pentagon and mark the center point ''O''. (This is the green circle in the diagram to the right).
#Choose a point ''A'' on the circle that will serve as one vertex of the pentagon. Draw a line through ''O'' and ''A''.
#Construct a line perpendicular to the line ''OA'' passing through ''O''. Mark its intersection with one side of the circle as the point ''B''.
#Construct the point ''C'' as the midpoint of ''O'' and ''B''.
#Draw a circle centered at ''C'' through the point ''A''. Mark its intersection with the line ''OB'' (inside the original circle) as the point ''D''.
#Draw a circle centered at ''A'' through the point ''D''. Mark its intersections with the original (green) circle as the points ''E'' and ''F''.
#Draw a circle centered at ''E'' through the point ''A''. Mark its other intersection with the original circle as the point ''G''.
#Draw a circle centered at ''F'' through the point ''A''. Mark its other intersection with the original circle as the point ''H''.
#Construct the regular pentagon ''AEGHF''.
After forming a regular convex pentagon, if you join the non-adjacent corners (drawing the diagonals of the pentagon), you obtain a [[pentagram]], with a smaller regular pentagon in the center. Or if you extend the sides until the non-adjacent ones meet, you obtain a larger pentagram.
A simple method of creating a regular pentagon from just a strip of paper is 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.
==Pentagons in nature==
<gallery>
Image:BhindiCutUp.jpg|Pentagonal cross-section of [[okra]] (also called "ladies fingers")
Image:Two okra flowers.JPG|The okra also has pentagonal flowers, each having five petals
Image:Sterappel dwarsdrsn.jpg|The [[gynoecium]] of an [[apple]] contains five carpels, arranged in a [[five-pointed star]]
Image:Carambola cut.jpg|Starfruit
Image:Belimbing.jpg|Belimbing
</gallery>
==See also==
* [[Exact trigonometric constants#36.C2.B0 - Pentagon|Trigonometric constants for a pentagon]]
* [[Pentagram]]
*[[The Pentagon]]
* [[Pentastar]]
* [[Dodecahedron]], a polyhedron whose regular form is composed of 12 pentagonal faces
==External links==
* {{MathWorld|title=Pentagon|urlname=Pentagon}}
*[http://www.opentutorial.com/Construct_a_pentagon How to construct a regular pentagon] using only compass and straightedge
*[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
*[http://www.geocities.com/robinhuiscool/Pentagon.html Nine constructions for the regular pentagon by Robin Hu]
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1056&bodyId=1245 Renaissance artists' approximate constructions of regular pentagons] at [http://mathdl.maa.org/convergence/1/ Convergence]
{{Polygons}}
[[Category:Polygons]]
[[ast:Pentágonu]]
[[az:Düzgün beşbucaqlı]]
[[ca:Pentàgon]]
[[cs:Pětiúhelník]]
[[da:Femkant]]
[[de:Fünfeck]]
[[el:Κανονικό_πεντάγωνο]]
[[es:Pentágono]]
[[eo:Kvinlatero]]
[[eu:Pentagono]]
[[fr:Pentagone (figure)]]
[[gl:Pentágono (xeometría)]]
[[ko:오각형]]
[[id:Segi lima]]
[[is:Fimmhyrningur]]
[[it:Pentagono (geometria)]]
[[he:מחומש]]
[[ht:Pentagòn]]
[[hu:Ötszög]]
[[nl:Vijfhoek]]
[[ja:五角形]]
[[no:Pentagon]]
[[nrm:Chîn-carres]]
[[km:បញ្ចកោណ]]
[[pl:Pięciokąt]]
[[pt:Pentágono]]
[[ro:Pentagon (geometrie)]]
[[ru:Пятиугольник]]
[[simple:Pentagon]]
[[sk:Päťuholník]]
[[sl:Petkotnik]]
[[sr:Петоугао]]
[[fi:Viisikulmio]]
[[sv:Pentagon]]
[[te:పంచభుజి]]
[[th:รูปห้าเหลี่ยม]]
[[tr:Beşgen]]
[[yi:פינפעק]]
[[zh:五边形]]