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]]