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regular polygon
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{{short description|Equiangular and equilateral polygon}}{| class=wikitable align="right" width="255"!bgcolor=#e7dcc3 colspan=2|Set of convex regular n-gons- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
60px60px60px60px60px60px60px60px60px60px60px60px60px60px60px60px60px60pxRegular polygons |
Edge (geometry) | s and Vertex (geometry)>vertices | n |
SchlÃ¤fli symbol | {n} |
Coxeterâ€“Dynkin diagram | {{CDD | n|node}} |
Point group | > | Dihedral symmetry>Dn, order 2n |
Dual polygon | Self-dual |
Area (with side length, s) | A = tfrac{1}{4}ns^2 cotleft(frac{pi}{n}right) |
Internal angle | (n - 2) times frac{180^circ}{n} |
Internal angle sum | left(n - 2right)times 180^circ |
Inscribed circle diameter | d_text{IC} = scotleft(frac{pi}{n}right) |
Circumscribed circle diameter | d_text{OC} = scscleft(frac{pi}{n}right) |
Properties | Convex polygon | , Cyclic polygon>cyclic, equilateral polygon | , isogonal figure>isogonal, isotoxal |
General properties
(File:regular star polygons.svg|thumb|300px|Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols)These properties apply to all regular polygons, whether convex or star.A regular n-sided polygon has rotational symmetry of order n.All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon.Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon.A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.Symmetry
The symmetry group of an n-sided regular polygon is dihedral group Dn (of order 2n): D2, D3, D4, ... It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side.Regular convex polygons
All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.An n-sided convex regular polygon is denoted by its SchlÃ¤fli symbol {n}. For n < 3, we have two degenerate cases:- Monogon {1}: Degenerate in ordinary space. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any abstract polygon.)
- Digon {2}; a "double line segment": Degenerate in ordinary space. (Some authorities do not regard the digon as a true polygon because of this.)
Angles
For a regular convex n-gon, each interior angle has a measure of:
left(1 - frac{2}{n}right)times 180,,, degrees, or equivalently frac{180(n - 2)}{n} degrees;
frac{(n - 2)pi}{n} radians; or
frac{(n - 2)}{2n} full turns,
and each exterior angle (i.e., supplementary to the interior angle) has a measure of tfrac{360}{n} degrees, with the sum of the exterior angles equal to 360 degrees or 2Ï€ radians or one full turn.As the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964Â°. As the number of sides increase, the internal angle can come very close to 180Â°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180Â°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with an infinite number of sides.Diagonals
For n > 2, the number of diagonals is tfrac{1}{2}n(n - 3); i.e., 0, 2, 5, 9, â€¦, for a triangle, square, pentagon, hexagon, â€¦ . The diagonals divide the polygon into 1, 4, 11, 24, â€¦ pieces {{oeis|A007678}}.For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n.Points in the plane
For a regular simple n-gon with circumradius R and distances di from an arbitrary point in the plane to the vertices, we havePark, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227-232.weblink
frac{sum_{i=1}^n d_i^4}{n} + 3R^4 = left(frac{sum_{i=1}^n d_i^2}{n} + R^2right)^2.
Interior points
For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothemJohnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).{{rp|p. 72}} (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n=3 case.Pickover, Clifford A, The Math Book, Sterling, 2009: p. 150Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", The College Mathematics Journal 37(5), 2006, pp. 390â€“391.Circumradius
missing image!
- PolygonParameters.png -
Regular pentagon (n = 5) with side s, circumradius R and apothem a
{{regular_polygon_side_count_graph.svg}}The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by
- PolygonParameters.png -
Regular pentagon (n = 5) with side s, circumradius R and apothem a
R = frac{s}{2 sinleft(frac{pi}{n}right)} = frac{a}{cosleft(frac{pi}{n} right)}
For constructible polygons, algebraic expressions for these relationships exist; see Bicentric polygon#Regular polygons.The sum of the perpendiculars from a regular n-gon's vertices to any line tangent to the circumcircle equals n times the circumradius.{{rp|p. 73}}The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR2 where R is the circumradius.{{rp|p.73}}The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR2 âˆ’ {{sfrac|ns2|4}}, where s is the side length and R is the circumradius.{{rp|p. 73}}{{Clear|left}}Dissections
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into tbinom{n}{2} or m(m-1)/2 parallelograms.These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m-cubes.Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi.The list {{OEIS2C|1=A006245}} gives the number of solutions for smaller polygons.{| class=wikitable|+ Example dissections for select even-sided regular polygons!2m!6!8!10!12!14!16!18!20!24!30!40!50 align=center valign=top!Image60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) | 60px) |
Area
The area A of a convex regular n-sided polygon having side s, circumradius R, apothem a, and perimeter p is given byWEB,weblink Math Open Reference, 4 Feb 2014, WEB,weblink Mathwords,
A = tfrac{1}{2}nsa = tfrac{1}{2}pa = tfrac{1}{4}ns^2cotleft(tfrac{pi}{n}right) = na^2tanleft(tfrac{pi}{n}right) = tfrac{1}{2}nR^2sinleft(tfrac{2pi}{n}right)
For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table:Results for R = 1 and a = 1 obtained with Maple, using function definition:f := proc (n)options operator, arrow;[
[convert(1/4*n*cot(Pi/n), radical), convert(1/4*n*cot(Pi/n), float)],
[convert(1/2*n*sin(2*Pi/n), radical), convert(1/2*n*sin(2*Pi/n), float), convert(1/2*n*sin(2*Pi/n)/Pi, float)],
[convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]
]end procThe expressions for n=16 are obtained by twice applying the tangent half-angle formula to tan(π/4) (Note that since cot x rightarrow 1/x as x rightarrow 0Trigonometric functions, the area when s=1 is tending to n^2/4pi as n grows large.){{Clear}}{| class=wikitable style="text-align:center;"! rowspan="2" | Number of sides! style="background:#FF8080" colspan="2" | Area when side s = 1! style="background:#80FF80" colspan="3" | Area when circumradius R = 1! style="background:#8080FF" colspan="3" | Area when apothem a = 1[convert(1/2*n*sin(2*Pi/n), radical), convert(1/2*n*sin(2*Pi/n), float), convert(1/2*n*sin(2*Pi/n)/Pi, float)],
[convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]
| | | tfrac{n}{2pi}sinleft(tfrac{2pi}{n}right) | | tfrac{n}{pi}tanleft(tfrac{pi}{n}right) |
Equilateral triangle>3 | tfrac{ sqrt{3} }{4} }} | 0.433012702 | tfrac{3sqrt{3} }{4} }} | 1.299038105 | 0.4134966714 | 3sqrt{3} }} | 5.196152424 | 1.653986686 |
Square>4 | | 1.000000000 | | 0.6366197722 | | 1.273239544 |
Regular pentagon>5 | tfrac{1}{4}sqrt{25 + 10sqrt{5} } }} | 1.720477401 | tfrac{5}{4}sqrt{tfrac{1}{2}left(5 + sqrt{5}right)} }} | 2.377641291 | 0.7568267288 | 5sqrt{5 - 2sqrt{5} } }} | 3.632712640 | 1.156328347 |
Regular hexagon>6 | tfrac{3sqrt{3} }{2} }} | 2.598076211 | tfrac{3sqrt{3} }{2} }} | 2.598076211 | 0.8269933428 | 2sqrt{3} }} | 3.464101616 | 1.102657791 |
Regular heptagon>7 | | 3.633912444 | | 0.8710264157 | | 1.073029735 |
Regular octagon>8 | 2 + 2sqrt{2} }} | 4.828427125 | 2sqrt{2} }} | 2.828427125 | 0.9003163160 | 8left(sqrt{2} - 1right)}} | 3.313708500 | 1.054786175 |
Regular enneagon>9 | | 6.181824194 | | 0.9207254290 | | 1.042697914 |
Regular decagon>10 | tfrac{5}{2}sqrt{5 + 2sqrt{5} } }} | 7.694208843 | tfrac{5}{2}sqrt{tfrac{1}{2}left(5 - sqrt{5}right)} }} | 2.938926262 | 0.9354892840 | 2sqrt{25 - 10sqrt{5} } }} | 3.249196963 | 1.034251515 |
Regular hendecagon>11 | | 9.365639907 | | 0.9465022440 | | 1.028106371 |
Regular dodecagon>12 | 6 + 3sqrt{3} }} | 11.19615242 | | 0.9549296586 | 12left(2 - sqrt{3} right)}} | 3.215390309 | 1.023490523 |
Regular tridecagon>13 | | 13.18576833 | | 0.9615188694 | | 1.019932427 |
Regular tetradecagon>14 | | 15.33450194 | | 0.9667663859 | | 1.017130161 |
Regular pentadecagon>15 | tfrac{15}{8}left(sqrt{15} + sqrt{3} + sqrt{2left(5 + sqrt{5} right)} right)}} | 17.64236291 | tfrac{15}{16}left(sqrt{15} + sqrt{3} - sqrt{10 - 2sqrt{5} } right)}} | 3.050524822 | 0.9710122088 | tfrac{15}{2}left(3sqrt{3} - sqrt{15} - sqrt{2left(25 - 11sqrt{5} right)} right)}} | 3.188348426 | 1.014882824 |
Regular hexadecagon>16 | 4 left(1 + sqrt{2} + sqrt{2 left(2 + sqrt{2} right)} right)}} | 20.10935797 | 4sqrt{2 - sqrt{2} } }} | 3.061467460 | 0.9744953584 | 16 left(1 + sqrt{2}right)left(sqrt{2 left(2 - sqrt{2} right)} - 1right)}} | 3.182597878 | 1.013052368 |
Regular heptadecagon>17 | | 22.73549190 | | 0.9773877456 | | 1.011541311 |
Regular octadecagon>18 | | 25.52076819 | | 0.9798155361 | | 1.010279181 |
Regular enneadecagon>19 | | 28.46518943 | | 0.9818729854 | | 1.009213984 |
Regular icosagon>20 | 5 left(1 + sqrt{5} + sqrt{5 + 2sqrt{5} } right) }} | 31.56875757 | tfrac{5}{2}left(sqrt{5} - 1right)}} | 3.090169944 | 0.9836316430 | 20 left(1 + sqrt{5} -sqrt{5 + 2sqrt{5} } right) }} | 3.167688806 | 1.008306663 |
Regular hectogon>100 | | 795.5128988 | | 0.9993421565 | | 1.000329117 |
Regular chiliagon>1000 | | 79577.20975 | | 0.9999934200 | | 1.000003290 |
Regular myriagon>10,000 | | 7957746.893 | | 0.9999999345 | | 1.000000033 |
Regular megagon>1,000,000 | | 79577471545 | | 1.000000000 | | 1.000000000 |
Constructible polygon
Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all.The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,{{rp|p. xi}} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).{{rp|pp. 49â€“50}} This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none).
(A Fermat prime is a prime number of the form 2^{(2^n)}+1.) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gaussâ€“Wantzel theorem.Equivalently, a regular n-gon is constructible if and only if the cosine of its common angle is a constructible numberâ€”that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.">{{anchor|Skew regular polygons}}Regular skew polygons{|classwikitable alignright width400 valign=top
160px)The cube contains a skew regular hexagon, seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis. | 240px)The zig-zagging side edges of a n-antiprism represent a regular skew 2n-gon, as shown in this 17-gonal antiprism. |
Regular star polygons{| class"wikitable floatright"|+ Regular star polygons
80px){5/2} | 80px){7/2} | 80px){7/3}... |
node_1 | rat | node}} |
Dihedral symmetry>Dihedral (Dp) |
TITLE=BEYOND MEASURE: A GUIDED TOUR THROUGH NATURE, MYTH, AND NUMBER | YEAR=2002 | ISBN= 978-981-02-4702-7,weblink |
- Pentagram â€“ {5/2}
- Heptagram â€“ {7/2} and {7/3}
- Octagram â€“ {8/3}
- Enneagram â€“ {9/2} and {9/4}
- Decagram â€“ {10/3}
- Hendecagram â€“ {11/2}, {11/3}, {11/4} and {11/5}
- Dodecagram â€“ {12/5}
- Tetragon â€“ {4/2}
- Hexagons â€“ {6/2}, {6/3}
- Octagons â€“ {8/2}, {8/4}
- Enneagon â€“ {9/3}
- Decagons â€“ {10/2}, {10/4}, and {10/5}
- Dodecagons â€“ {12/2}, {12/3}, {12/4}, and {12/6}
160px) | 160px) |
- For much of the 20th century (see for example {{harvtxt|Coxeter|1948}}), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbors two steps away, to obtain the regular compound of two triangles, or hexagram. {{paragraph}}Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so the hexagram is represented as {6}[2{3}]{6}.Regular polytopes, p.95 More compactly Coxeter also writes 2{n/2}, like 2{3} for a hexagram as compound as alternations of regular even-sided polygons, with italics on the leading factor to differentiate it from the coinciding interpretation.Coxeter, The Densities of the Regular Polytopes II, 1932, p.53
- Many modern geometers, such as GrÃ¼nbaum (2003), regard this as incorrect. They take the /2 to indicate moving two places around the {6} at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories of abstract polytopes, but it also more closely copies the way in which Poinsot (1809) created his star polygons â€“ by taking a single length of wire and bending it at successive points through the same angle until the figure closed.
Duality of regular polygons
{{see also|Self-dual polyhedra}}All regular polygons are self-dual to congruency, and for odd n they are self-dual to identity.In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.Regular polygons as faces of polyhedra
A uniform polyhedron has regular polygons as faces, such that for every two vertices there is an isometry mapping one into the other (just as there is for a regular polygon).A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex.A regular polyhedron is a uniform polyhedron which has just one kind of face.The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids.A polyhedron having regular triangles as faces is called a deltahedron.See also
- Euclidean tilings by convex regular polygons
- Platonic solid
- Apeirogon â€“ An infinite-sided polygon can also be regular, {âˆž}.
- List of regular polytopes and compounds
- Equilateral polygon
- Carlyle circle
Notes
{{Reflist}}References
- JOURNAL, Coxeter, H.S.M., Coxeter, Regular Polytopes, Methuen and Co., 1948, harv,
- GrÃ¼nbaum, B.; Are your polyhedra the same as my polyhedra?, Discrete and comput. geom: the Goodman-Pollack festschrift, Ed. Aronov et al., Springer (2003), pp. 461â€“488.
- Poinsot, L.; Memoire sur les polygones et polyÃ¨dres. J. de l'Ã‰cole Polytechnique 9 (1810), pp. 16â€“48.
External links
- {{mathworld |urlname=RegularPolygon |title=Regular polygon}}
- Regular Polygon description With interactive animation
- Incircle of a Regular Polygon With interactive animation
- Area of a Regular Polygon Three different formulae, with interactive animation
- Renaissance artists' constructions of regular polygons at Convergence
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