Decagon - wikipedia Decagon shape with ten sides Regular decagon A regular decagon Type Regular polygon Edges and vertices 10 Schläfli symbol ( 10 ), t ( 5 ) Coxeter diagram Symmetry group Dihedral ( D ), order 2 × 10 Internal angle ( degrees ) 144 ° Dual polygon Self Properties Convex, cyclic, equilateral, isogonal, isotoxal In geometry, a decagon is a ten - sided polygon or 10 - gon. Contents 1 Regular decagon 1.1 Area 1.2 Sides 1.3 Construction 1.4 Nonconvex regular decagon 2 The golden ratio in decagon 3 Symmetry 4 Dissection 5 Skew decagon 5.1 Petrie polygons 6 See also 7 References 8 External links Regular decagon A regular decagon has all sides of equal length and each internal angle will always be equal to 144 °. Its Schläfli symbol is ( 10 ) and can also be constructed as a truncated pentagon, t ( 5 ), a quasiregular decagon alternating two types of edges. Area The area of a regular decagon of side length a is given by : A = 5 2 a 2 cot ⁡ ( π 10 ) = 5 2 a 2 5 + 2 5 ≃ 7.694208843 a 2 ( displaystyle A = ( \frac ( 5 ) ( 2 ) ) a ^ ( 2 ) cot left ( ( \frac ( pi ) ( 10 ) ) \right ) = ( \frac ( 5 ) ( 2 ) ) a ^ ( 2 ) ( sqrt ( 5 + 2 ( sqrt ( 5 ) ) ) ) simeq 7.694208843 , a ^ ( 2 ) ) In terms of the apothem r ( see also inscribed figure ), the area is : A = 10 tan ⁡ ( π 10 ) r 2 = 5 r 2 5 − 2 5 ≃ 3.249196963 r 2 ( displaystyle A = 10 \tan left ( ( \frac ( pi ) ( 10 ) ) \right ) r ^ ( 2 ) = 5r ^ ( 2 ) ( sqrt ( 5 - 2 ( sqrt ( 5 ) ) ) ) simeq 3.249196963 , r ^ ( 2 ) ) In terms of the circumradius R, the area is : A = 5 2 sin ⁡ ( π 5 ) R 2 = 5 2 R 2 5 − 5 2 ≃ 2.938926262 R 2 ( displaystyle A = ( \frac ( 5 ) ( 2 ) ) sin left ( ( \frac ( pi ) ( 5 ) ) \right ) R ^ ( 2 ) = ( \frac ( 5 ) ( 2 ) ) R ^ ( 2 ) ( sqrt ( \frac ( 5 - ( sqrt ( 5 ) ) ) ( 2 ) ) ) simeq 2.938926262 , R ^ ( 2 ) ) An alternative formula is A = 2.5 d a ( displaystyle A = 2.5 da ) where d is the distance between parallel sides, or the height when the decagon stands on one side as base, or the diameter of the decagon 's inscribed circle. By simple trigonometry, d = 2 a ( cos ⁡ 3 π 10 + cos ⁡ π 10 ), ( displaystyle d = 2a left ( cos ( \tfrac ( 3 pi ) ( 10 ) ) + cos ( \tfrac ( pi ) ( 10 ) ) \right ), ) and it can be written algebraically as d = a 5 + 2 5. ( displaystyle d = a ( sqrt ( 5 + 2 ( sqrt ( 5 ) ) ) ). ) Sides The side of a regular decagon inscribed in a unit circle is − 1 + 5 2 = 1 φ ( displaystyle ( \tfrac ( - 1 + ( sqrt ( 5 ) ) ) ( 2 ) ) = ( \tfrac ( 1 ) ( phi ) ) ), where φ is the golden ratio, 1 + 5 2 ( displaystyle ( \tfrac ( 1 + ( sqrt ( 5 ) ) ) ( 2 ) ) ). Construction As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge - bisection of a regular pentagon. Construction of decagon Construction of pentagon An alternative ( but similar ) method is as follows : Construct a pentagon in a circle by one of the methods shown in constructing a pentagon. Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon. Nonconvex regular decagon This tiling by golden triangles, a regular pentagon, contains a stellation of regular decagon, the Schäfli symbol of which is ( 10 / 3 ). The length ratio of two inequal edges of a golden triangle is the golden ratio, denoted by Φ, ( displaystyle ( \text ( by ) ) Phi ( \text (, ) ) ) or its multiplicative inverse : Φ − 1 = 1 Φ = 2 cos ⁡ 72 ∘ = 1 2 cos ⁡ 36 ∘ = 5 − 1 2. ( displaystyle Phi - 1 = ( \frac ( 1 ) ( Phi ) ) = 2 , cos 72 , ^ ( circ ) = ( \frac ( 1 ) ( , 2 , cos 36 , ^ ( circ ) ) ) = ( \frac ( , ( sqrt ( 5 ) ) - 1 , ) ( 2 ) ) ( \text (. ) ) ) So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this star polygon. The golden ratio in decagon Both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element. In the construction with given circumcircle the circular arc around G with radius GE produces the segment AH, whose division corresponds to the golden ratio. A M  ̄ M H  ̄ = A H  ̄ A M  ̄ = 1 + 5 2 = Φ ≈ 1.618. ( displaystyle ( \frac ( overline ( AM ) ) ( overline ( MH ) ) ) = ( \frac ( overline ( AH ) ) ( overline ( AM ) ) ) = ( \frac ( 1 + ( sqrt ( 5 ) ) ) ( 2 ) ) = Phi approx 1.618 ( \text (. ) ) ) In the construction with given side length the circular arc around D with radius DA produces the segment E F, whose division corresponds to the golden ratio. E 1 E 10  ̄ E 1 F  ̄ = E 10 F  ̄ E 1 E 10  ̄ = R a = 1 + 5 2 = Φ ≈ 1.618. ( displaystyle ( \frac ( overline ( E_ ( 1 ) E_ ( 10 ) ) ) ( overline ( E_ ( 1 ) F ) ) ) = ( \frac ( overline ( E_ ( 10 ) F ) ) ( overline ( E_ ( 1 ) E_ ( 10 ) ) ) ) = ( \frac ( R ) ( a ) ) = ( \frac ( 1 + ( sqrt ( 5 ) ) ) ( 2 ) ) = Phi approx 1.618 ( \text (. ) ) ) Decagon with given circumcircle, animation Decagon with a given side length, animation Symmetry Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center. The regular decagon has Dih symmetry, order 20. There are 3 subgroup dihedral symmetries : Dih, Dih, and Dih, and 4 cyclic group symmetries : Z, Z, Z, and Z. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. Full symmetry of the regular form is r20 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 g10 subgroup has no degrees of freedom but can seen as directed edges. The highest symmetry irregular decagons are d10, a isogonal decagon constructed by five mirrors which can alternate long and short edges, and p10, an isotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular decagon. Dissection 10 - cube projection 40 rhomb dissection Coxeter states that every zonogon ( a 2m - gon whose opposite sides are parallel and of equal length ) can be dissected into m ( m - 1 ) / 2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular decagon, m = 5, and it can be divided into 10 rhombs, with examples shown below. This decomposition can be seen as 10 of 80 faces in a Petrie polygon projection plane of the 5 - cube. A dissection is based on 10 of 30 faces of the rhombic triacontahedron. The list A006245 defines the number of solutions as 62, with 2 orientations for the first symmetric form, and 10 orientations for the other 6. Regular decagon dissected into 10 rhombi 5 - cube Skew decagon 3 regular skew zig - zag decagons ( 5 ) # ( ) ( 5 / 2 ) # ( ) ( 5 / 3 ) # ( ) A regular skew decagon is seen as zig - zagging edges of a pentagonal antiprism, a pentagrammic antiprism, and a pentagrammic crossed - antiprism. A skew decagon is a skew polygon with 10 vertices and edges but not existing on the same plane. The interior of such an decagon is not generally defined. A skew zig - zag decagon has vertices alternating between two parallel planes. A regular skew decagon is vertex - transitive with equal edge lengths. In 3 - dimensions it will be a zig - zag skew decagon and can be seen in the vertices and side edges of a pentagonal antiprism, pentagrammic antiprism, and pentagrammic crossed - antiprism with the same D, ( 2, 10 ) symmetry, order 20. These can also be seen in these 4 convex polyhedra with icosahedral symmetry. The polygons on the perimeter of these projections are regular skew decagons. Orthogonal projections of polyhedra on 5-fold axes Dodecahedron Icosahedron Icosidodecahedron Rhombic triacontahedron Petrie polygons The regular skew decagon is the Petrie polygon for many higher - dimensional polytopes, shown in these orthogonal projections in various Coxeter planes : The number of sides in the Petrie polygon is equal to the Coxeter number, h, for each symmetry family. 9 - simplex 5 - orthoplex 5 - cube See also Decagonal number and centered decagonal number, figurate numbers modeled on the decagon Decagram, a star polygon with the same vertex positions as the regular decagon References ^ Jump up to : Sidebotham, Thomas H. ( 2003 ), The A to Z of Mathematics : A Basic Guide, John Wiley & Sons, p. 146, ISBN 9780471461630. Jump up ^ Wenninger, Magnus J. ( 1974 ), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595. Jump up ^ The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge, 1850, p. 59. Note that this source uses a as the edge length and gives the argument of the cotangent as an angle in degrees rather than in radians. ^ Jump up to : Ludlow, Henry H. ( 1904 ), Geometric Construction of the Regular Decagon and Pentagon Inscribed in a Circle, The Open Court Publishing Co... ^ Jump up to : Green, Henry ( 1861 ), Euclid 's Plane Geometry, Books III -- VI, Practically Applied, or Gradations in Euclid, Part II, London : Simpkin, Marshall, & CO., p. 116. Retrieved 10 February 2016. ^ Jump up to : ( in German ) Köller, Jürgen ( 2005 ), Regelmäßiges Zehneck, → 3. Section `` Formeln, Ist die Seite a gegeben... ''. Retrieved 10 February 2016. Jump up ^ 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 ) Jump up ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p. 141 Jump up ^ Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223 - 226. External links Weisstein, Eric W. `` Decagon ''. MathWorld. Definition and properties of a decagon With interactive animation Polygons ( List ) 1 -- 10 sides Monogon Digon Triangle Equilateral Isosceles Right Acute / Obtuse Quadrilateral Square Rectangle Rhombus Parallelogram Trapezoid Isosceles trapezoid Kite Pentagon Hexagon Heptagon Octagon Nonagon ( Enneagon ) Decagon 11 -- 20 sides Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon 21 -- 100 sides ( selected ) Icosidigon ( 22 ) Icositetragon ( 24 ) Icosihexagon ( 26 ) Icosioctagon ( 28 ) Triacontagon ( 30 ) Triacontadigon ( 32 ) Triacontatetragon ( 34 ) Tetracontagon ( 40 ) Tetracontadigon ( 42 ) Tetracontaoctagon ( 48 ) Pentacontagon ( 50 ) Hexacontagon ( 60 ) Hexacontatetragon ( 64 ) Heptacontagon ( 70 ) Octacontagon ( 80 ) Enneacontagon ( 90 ) Enneacontahexagon ( 96 ) Hectogon ( 100 ) > 100 sides 120 - gon 257 - gon 360 - gon Chiliagon ( 1000 ) Myriagon ( 10000 ) 65537 - gon Megagon ( 1000000 ) Apeirogon ( ∞ ) Star polygons ( 5 -- 12 sides ) Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram Others Pseudogon Skew polygon Skew apeirogon Classes Concave Convex Cyclic Equiangular Equilateral Isogonal Isotoxal Regular Simple Tangential Retrieved from `` https://en.wikipedia.org/w/index.php?title=Decagon&oldid=863100275 '' Categories : Polygons 10 ( number ) Constructible polygons Hidden categories : Articles with German - language external links Articles with short description Talk Contents About Wikipedia Asturianu Башҡортса Български Català Čeština Deutsch Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Gaelg Galego 客家 語 / Hak - kâ - ngî 한국어 Հայերեն Ido Bahasa Indonesia Italiano ქართული Kreyòl ayisyen Latviešu Lumbaart Magyar Македонски Bahasa Melayu Nederlands 日本 語 Norsk Norsk nynorsk پښتو Português Română Русский Scots Simple English Slovenščina کوردی Српски / srpski Suomi Svenska தமிழ் ไทย Türkçe Українська Tiếng Việt 粵語 中文 42 more Edit links This page was last edited on 8 October 2018, at 18 : 25 ( UTC ). 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Question: what do you call a ten sided shape
Answer: decagon