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| {{two other uses|the geometric figure|the headquarters of the United States Department of Defense|The Pentagon}}
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| {{Odd polygon db|Odd polygon stat table|p5}}
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| In [[geometry]], a '''pentagon''' (from ''pente'' and ''gonia'', which is [[Greek language|Greek]] for ''five'' and ''angle'') is any five-sided [[polygon]]. A pentagon may be simple or self-intersecting. The sum of the [[internal angle]]s in a [[simple polygon|simple]] pentagon is 540°. A [[pentagram]] is an example of a self-intersecting pentagon.
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| ==Regular pentagons==
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| In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has five lines of [[reflectional symmetry]], and [[rotational symmetry]] of order 5 (through 72°, 144°, 216° and 288°). Its [[Schläfli symbol]] is {5}. The [[diagonal]]s of a regular pentagon are in [[golden ratio]] to its sides.
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| The area of a regular convex pentagon with side length ''t'' is given by
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| :<math>A = \frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4} = \frac{5t^2 \tan(54^\circ)}{4} \approx 1.720477401 t^2.</math>
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| 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]].
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| When a regular pentagon is [[Inscribed figure|inscribed]] in a circle with radius ''R'', its edge length ''t'' is given by the expression
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| :<math>t = R\ {\sqrt { \frac {5-\sqrt{5}}{2}} } = 2R\sin 36^\circ = 2R\sin\frac{\pi}{5} \approx 1.17557050458 R.</math>
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| ===Derivation of the area formula===
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| The area of any regular polygon is:
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| :<math>A = \frac{1}{2}Pa</math>
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| where ''P'' is the perimeter of the polygon, ''a'' is the [[apothem]]. One can then substitute the respective values for ''P'' and ''a'', which makes the formula:
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| :<math>A = \frac{1}{2} \times \frac{5t}{1} \times \frac{t\tan(54^\circ)}{2}</math>
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| with ''t'' as the given side length. Then we can then rearrange the formula as:
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| :<math>A = \frac{1}{2} \times \frac{5t^2\tan(54^\circ)}{2}</math>
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| and then, we combine the two terms to get the final formula, which is:
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| :<math>A = \frac{5t^2\tan(54^\circ)}{4}.</math>
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| ===Derivation of the diagonal length formula===
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| The diagonals of a regular pentagon (hereby represented by ''D'') can be calculated based upon the golden ratio φ and the known side ''T'' (see discussion of the pentagon in [[Golden ratio#The side and diagonal of the regular pentagon|Golden ratio]]):
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| :<math>\frac {D}{T} = \varphi = \frac {1+ \sqrt {5} }{2} \ , </math>
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| Accordingly:
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| :<math>D = T \times \varphi \ = R\ {\sqrt { \frac {5+\sqrt{5}}{2}} } = 2R\cos 18^\circ = 2R\cos\frac{\pi}{10} \approx 1.90211303259 R.</math>
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| ===Chords from the circumscribing circle to the vertices===
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| If a regular pentagon with successive vertices A, B, C, D, E is inscribed in a circle, and if P is any point on that circle between points B and C, then PA + PD = PB + PC + PE.
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| ==Construction of a regular pentagon==
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| A variety of methods are known for constructing a regular pentagon. Some are discussed below.
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| ===Euclid's methods===
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| 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=http://books.google.com/books?id=ABLtD3IE_RQC&pg=PA6 |page=6 |isbn=0-387-98276-0 |year=1998 |publisher=Springer}}</ref>
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| ===Richmond's method===
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| One method to construct a regular pentagon in a given circle is described by Richmond<ref name=Richmond>{{cite web |url=http://mathworld.wolfram.com/Pentagon.html |title=Pentagon |author=Herbert W Richmond |year=1893}}</ref> and further discussed in Cromwell's "Polyhedra."<ref>{{cite book |title=Polyhedra |author=Peter R. Cromwell |page=63 |url=http://books.google.com/books?id=OJowej1QWpoC&pg=PA63 |isbn=0-521-66405-5}}</ref>
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| ===Verification===
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| [[File:Richmond pentagon 1.PNG|160px|thumb]]
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| [[File:Richmond Pentagon 2.PNG|160px|thumb]]
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| The top panel describes the construction used in the animation above 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.
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| 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]]:
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| :<math>\tan ( \phi/2) = \frac{1-\cos(\phi)}{\sin (\phi)} \ ,</math>
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| where cosine and sine of ''ϕ'' are known from the larger triangle. The result is:
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| :<math>h = \frac{\sqrt 5 - 1}{4} \ .</math>
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| With this side known, attention turns to the lower diagram to find the side ''s'' of the regular pentagon. First, side ''a'' of the right-hand triangle is found using Pythagoras' theorem again:
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| :<math>a^2 = 1-h^2 \ ; \ a = \frac{1}{2}\sqrt { \frac {5+\sqrt 5}{2}} \ .</math>
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| Then ''s'' is found using Pythagoras' theorem and the left-hand triangle as:
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| :<math>s^2 = (1-h)^2 + a^2 = (1-h)^2 + 1-h^2 = 1-2h+h^2 + 1-h^2 = 2-2h=2-2\left(\frac{\sqrt 5 - 1}{4}\right) \ </math>
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| ::<math>=\frac {5-\sqrt 5}{2} \ .</math>
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| The side ''s'' is therefore:
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| :<math>s = \sqrt{ \frac {5-\sqrt 5}{2}} \ ,</math>
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| a well established result.<ref name=Touton>This result agrees with {{cite book |chapter=Exercise 175 |page=302 |author=Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton |url=http://books.google.com/books?id=eOdHAAAAIAAJ&pg=PA302 |title=Plane geometry |year=1920 |publisher=Ginn & Co.}}</ref> Consequently, this construction of the pentagon is valid.
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| ===Alternative method===
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| An alternative method is this:
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| [[File:pentagon-construction.svg|thumb|Constructing a pentagon]]
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| #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).
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| #Choose a point ''A'' on the circle that will serve as one vertex of the pentagon. Draw a line through ''O'' and ''A''.
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| #Construct a line perpendicular to the line ''OA'' passing through ''O''. Mark its intersection with one side of the circle as the point ''B''.
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| #Construct the point ''C'' as the midpoint of the line ''OB''.
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| #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''.
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| #Draw a circle centered at ''A'' through the point ''D''. Mark its intersections with the original (green) circle as the points ''E'' and ''F''.
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| #Draw a circle centered at ''E'' through the point ''A''. Mark its other intersection with the original circle as the point ''G''.
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| #Draw a circle centered at ''F'' through the point ''A''. Mark its other intersection with the original circle as the point ''H''.
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| #Construct the regular pentagon ''AEGHF''.
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| [[File:Regular Pentagon Inscribed in a Circle 240px.gif|center|frame|Animation that is almost the same as this alternative method]]
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| ===Carlyle circles===
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| ''See main article'': [[Carlyle circle]]
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| [[File:Regular Pentagon Using Carlyle Circle.gif|thumb|Method using Carlyle circles]]
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| 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=http://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 |title=Carlyle Circles and the Lemoine Simplicity of Polygon Constructions |author=Duane W DeTemple |journal=The American Mathematical Monthly |volume=98 |issue=2 |year=1991 |pages=97–108 |url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf}} [http://www.jstor.org/stable/2323939 JSTOR link]</ref>
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| #Draw a [[circle]] in which to inscribe the pentagon and mark the center point ''O''.
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| #Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point ''B''.
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| #Construct a vertical line through the center. Mark one intersection with the circle as point ''A''.
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| #Construct the point ''M'' as the midpoint of ''O'' and ''B''.
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| #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''.
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| #Draw a circle of radius ''OA'' and center ''W''. It intersects the original circle at two of the vertices of the pentagon.
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| #Draw a circle of radius ''OA'' and center ''V''. It intersects the original circle at two of the vertices of the pentagon.
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| #The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
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| Steps 6-8 are equivalent to the following version, shown in the animation:
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| <nowiki> </nowiki> 6a. Construct point F as the midpoint of O and W.
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| <nowiki> </nowiki> 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.
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| <nowiki> </nowiki> 8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.
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| ===Direct method===
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| A direct method using degrees follows:
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| #Draw a circle and choose a point to be the pentagon's (e.g. top center)
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| #Choose a point ''A'' on the circle that will serve as one vertex of the pentagon. Draw a line through ''O'' and ''A''.
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| #Draw a guideline through it and the circle's center
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| #Draw lines at 54° (from the guideline) intersecting the pentagon's point
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| #Where those intersect the circle, draw lines at 18° (from parallels to the guideline)
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| #Join where they intersect the circle
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| After forming a regular convex pentagon, if one joins the non-adjacent corners (drawing the diagonals of the pentagon), one obtains a [[pentagram]], with a smaller regular pentagon in the center. Or if one extends the sides until the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles.
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| ===Simple methods===
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| [[File:Knot.jpg|100px|thumb|Overhand knot of a paper strip]]
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| *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.
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| *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.
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| ==Cyclic pentagons==
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| 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>Robbins, D. P. "Areas of Polygons Inscribed in a Circle." ''Discr. Comput. Geom.'' 12, 223-236, 1994.</ref><ref>Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523-530, 1995.</ref>
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| There exist cyclic pentagons with rational sides and rational area; these are called [[Robbins pentagon]]s. In a Robbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all the diagonals must be rational.<ref>*{{citation
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| | last1 = Buchholz | first1 = Ralph H.
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| | last2 = MacDougall | first2 = James A.
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| | doi = 10.1016/j.jnt.2007.05.005
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| | issue = 1
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| | journal = [[Journal of Number Theory]]
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| | mr = 2382768
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| | pages = 17–48
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| | title = Cyclic polygons with rational sides and area
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| | url = http://docserver.carma.newcastle.edu.au/785/
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| | volume = 128
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| | year = 2008}}.</ref>
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| ==Graphs==
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| The K<sub>5</sub> [[complete graph]] is often drawn as a ''regular pentagon'' with all 10 edges connected. This graph also represents an [[orthographic projection]] of the 5 vertices and 10 edges of the [[5-cell]]. The [[rectified 5-cell]], with vertices at the mid-edges of the 5-cell is projected inside a pentagon.
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| {| class=wikitable
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| |- align=center
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| |[[File:4-simplex t0.svg|150px]]<br>[[5-cell]] (4D)
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| |[[File:4-simplex t1.svg|150px]]<br>[[Rectified 5-cell]] (4D)
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| |}
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| ==Examples of pentagons==
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| ===Plants===
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| <gallery>
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| Image:BhindiCutUp.jpg|Pentagonal cross-section of [[okra]].
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| Image:Morning_Glory_Flower.jpg|[[Morning glories]], like many other flowers, have a pentagonal shape.
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| Image:Sterappel dwarsdrsn.jpg|The [[gynoecium]] of an [[apple]] contains five carpels, arranged in a [[five-pointed star]]
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| Image:Carambola Starfruit.jpg|[[carambola|Starfruit]] is another fruit with fivefold symmetry.
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| </gallery>
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| ===Animals===
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| <gallery>
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| Image:Cervena_morska_hviezdica.jpg|A [[sea star]]. Many [[echinoderms]] have fivefold radial symmetry.
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| Image:Haeckel_Ophiodea.jpg|An illustration of [[brittle stars]], also echinoderms with a pentagonal shape.
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| </gallery>
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| ===Artificial===
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| <gallery>
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| Image:The Pentagon January 2008.jpg|[[The Pentagon]], headquarters of the [[United States Department of Defense]].
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| Image:Home base of baseball field in Třebíč, Czech Republic.jpg|[[Home plate]] of a [[baseball field]]
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| </gallery>
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| ==Pentagons in tiling==
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| {{Main|Pentagon tiling}}
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| [[File:2-d pentagon packing dual.svg|thumb|right|The best known packing of equal-sized regular pentagons on a plane is a double lattice structure which covers 92.131% of the plane.]]
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| A pentagon cannot appear in any tiling made by [[regular polygon]]s. To prove a pentagon cannot form a [[regular tiling]] (one in which all faces are congruent), observe that {{nowrap|360 / 108 {{=}} 3{{frac|1|3}}}}, which is not a whole number. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:
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| There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 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.
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| ==See also==
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| *[[Associahedron]]; A pentagon is an order-4 associahedron
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| *[[Dodecahedron]], a polyhedron whose regular form is composed of 12 pentagonal faces
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| *[[Exact trigonometric constants#36°: pentagon|Trigonometric constants for a pentagon]]
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| *[[Golden ratio]]
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| *[[List of geometric shapes]]
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| *[[Pentagonal number]]s
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| *[[Pentagram]]
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| *[[Pentagram map]]
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| *[[Pentastar]], the Chrysler logo
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| *[[Pythagoras' theorem]]
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| ==In-line notes and references==
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| <references/>
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| ==External links==
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| {{wiktionary|pentagon}}
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| *{{MathWorld|title=Pentagon|urlname=Pentagon}}
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| *[http://www.mathopenref.com/constinpentagon.html Animated demonstration] constructing an inscribed pentagon with compass and straightedge.
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| *[http://www.opentutorial.com/Construct_a_pentagon How to construct a regular pentagon] with only a compass and straightedge.
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| *[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
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| *[http://www.mathopenref.com/pentagon.html Definition and properties of the pentagon], with interactive animation
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| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1056&bodyId=1245 Renaissance artists' approximate constructions of regular pentagons]
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| *[http://whistleralley.com/polyhedra/pentagon.htm Pentagon.] How to calculate various dimensions of regular pentagons.
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| {{Polygons}}
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| [[Category:Polygons]]
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| [[Category:Elementary shapes]]
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