View source for Equiangular polygon

{{Short description|Polygon with equally angled vertices}}
{| class=wikitable align=right width=360
|+ Example equiangular polygons
!Direct||Indirect||Skew
|- valign=top
|[[File:equiangular_rectangle2x3.svg|100px]]<BR>A [[rectangle]], {{angbr|4}}, is a convex ''direct equiangular polygon'', containing four 90° internal angles.
|[[File:Tetromino-l2.svg|100px]]<BR>A concave ''indirect equiangular polygon'', {{angbr|6-2}}, like this hexagon, counterclockwise, has five left turns and one right turn, like this [[tetromino]].
|[[File:Isogonal skew octagon on cube.png|120px]]<BR>A [[skew polygon]] has equal angles off a plane, like this skew octagon alternating red and blue edges on a [[cube]].
|-
!Direct||Indirect||Counter-turned
|- valign=top
|[[File:Spirolateral 1-2-2-3-3-2-2-1 90.svg|120px]]<BR>A [[density (polygon)|multi-turning]] equiangular polygon can be direct, like this octagon, {{angbr|8/2}}, has 8 90° turns, totaling 720°.
|[[File:Equiangular pentagon2 60.svg|120px]]<BR>A concave indirect equiangular polygon, {{angbr|5-2}}, counterclockwise has 4 left turns and one right turn.<BR>(-1.2.4.3.2)<sub>60°</sub>
|[[File:Spirolateral 112-1-1-2 90.svg|120px]]<BR>An indirect equiangular hexagon, {{angbr|6-6}}<sub>90°</sub> with 3 left turns, 3 right turns, totaling 0°.
|}
In [[Euclidean geometry]], an '''equiangular polygon''' is a [[polygon]] whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also [[equilateral polygon|equilateral]]) then it is a [[regular polygon]]. [[Isogonal polygon]]s are equiangular polygons which alternate two edge lengths.

For clarity, a '''planar equiangular polygon''' can be called ''direct'' or ''indirect''. A '''direct equiangular polygon''' has all angles turning in the same direction in a plane and can include multiple [[turn (angle)|turns]]. '''Convex equiangular polygons''' are always direct. An '''indirect equiangular polygon''' can include angles turning right or left in any combination. A '''[[skew polygon|skew]] equiangular polygon''' may be [[isogonal figure|isogonal]], but can't be considered direct since it is nonplanar.

A [[spirolateral]] ''n''<sub>&theta;</sub> is a special case of an ''equiangular polygon'' with a set of ''n'' integer edge lengths repeating sequence until returning to the start, with vertex internal angles &theta;.

== Construction==
An ''equiangular polygon'' can be constructed from a [[regular polygon]] or [[regular star polygon]] where edges are extended as infinite [[Line (geometry)|line]]s. Each edges can be independently moved perpendicular to the line's direction. Vertices represent the intersection point between pairs of neighboring line. Each moved line adjusts its edge-length and the lengths of its two neighboring edges.<ref>Marius Munteanu, Laura Munteanu, [https://www.researchgate.net/publication/271296486_Rational_Equiangular_Polygons Rational Equiangular Polygons] Applied Mathematics, Vol.4 No.10, October 2013</ref> If edges are reduced to zero length, the polygon becomes degenerate, or if reduced to ''negative'' lengths, this will reverse the internal and external angles.

For an even-sided direct equiangular polygon, with internal angles &theta;°, moving alternate edges can invert all vertices into [[supplementary angle]]s, 180-&theta;°. Odd-sided direct equiangular polygons can only be partially inverted, leaving a mixture of supplementary angles.

Every equiangular polygon can be adjusted in proportions by this construction and still preserve equiangular status.

{| class=wikitable width=320
|[[File:Equiangular hexagon-example.png|160px]]<BR>This convex ''direct equiangular hexagon'', {{angbr|6}}, is bounded by 6 [[Line (geometry)|lines]] with 60° angle between. Each line can be moved perpendicular to its direction.
|[[File:Indirect equiangular polygon.png|160px]]<BR>This concave ''indirect equiangular hexagon'', {{angbr|6-2}}, is also bounded by 6 lines with 90° angle between, each line moved independently, moving vertices as new intersections.
|}

==Equiangular polygon theorem==
For a ''convex equiangular'' ''p''-gon, each [[internal angle]] is 180(1&minus;2/''p'')°; this is the ''equiangular polygon theorem''.

For a direct equiangular ''p''/''q'' [[star polygon]], [[density (geometry)|density]] ''q'', each internal angle is 180(1&minus;2''q''/''p'')°, with {{math|1 < 2''q'' < ''p''}}. For {{math|1=''w'' = gcd(''p'',''q'') > 1}}, this represents a ''w''-wound {{sfrac|''p''/''w''|''q''/''w''}} star polygon, which is degenerate for the regular case.

A concave ''indirect equiangular'' {{math|(''p''<sub>''r''</sub>+''p''<sub>''l''</sub>)}}-gon, with {{math|''p''<sub>''r''</sub>}} right turn vertices and {{math|''p''<sub>''l''</sub>}} left turn vertices, will have internal angles of {{math|180(1&minus;2/{{abs|''p''<sub>''r''</sub>&minus;''p''<sub>''l''</sub>}}))°}}, regardless of their sequence. An ''indirect star equiangular'' {{math|(''p''<sub>''r''</sub>+''p''<sub>''l''</sub>)}}-gon, with {{math|''p''<sub>''r''</sub>}} right turn vertices and {{math|''p''<sub>''l''</sub>}} left turn vertices and ''q'' total [[turn (angle)|turns]], will have internal angles of {{math|180(1&minus;2''q''/{{abs|''p''<sub>''r''</sub>&minus;''p''<sub>''l''</sub>}}))°}}, regardless of their sequence. An equiangular polygon with the same number of right and left turns has zero total turns, and has no constraints on its angles.

== Notation ==
Every direct equiangular ''p''-gon can be given a notation {{angbr|''p''}} or {{angbr|''p''/''q''}}, like [[regular polygon]]s {''p''} and [[regular star polygon]]s {''p''/''q''}, containing ''p'' vertices, and stars having [[Density (polytope)#Polygons|density]] ''q''.

Convex equiangular ''p''-gons {{angbr|''p''}} have internal angles 180(1&minus;2/''p'')°, while direct star equiangular polygons, {{angbr|''p''/''q''}}, have internal angles 180(1&minus;2''q''/''p'')°.

A concave indirect equiangular ''p''-gon can be given the notation {{angbr|''p''&minus;2''c''}}, with ''c'' counter-turn vertices. For example, {{angbr|6&minus;2}} is a hexagon with 90° internal angles of the difference, {{angbr|4}}, 1 counter-turned vertex. A multiturn indirect equilateral ''p''-gon can be given the notation {{angbr|<sup>''p''&minus;2''c''</sup>/<sub>''q''</sub>}}  with ''c'' counter turn vertices, and ''q'' total [[turn (angle)|turns]]. An equiangular polygon <''p''&minus;''p''> is a ''p''-gon with undefined internal angles {{mvar|&theta;}}, but can be expressed explicitly as {{angbr|''p''&minus;''p''}}<sub>''&theta;''</sub>.

==Other properties==
[[Viviani's theorem]] holds for equiangular polygons:<ref name="ref1">[https://arxiv.org/abs/0903.0753v3 Elias Abboud "On Viviani's Theorem and its Extensions"] pp. 2, 11</ref>
:''The sum of distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point, and is that polygon's invariant.''

A [[cyclic polygon]] is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus if ''n'' is odd, a cyclic polygon is equiangular if and only if it is regular.<ref>De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", [[Mathematical Gazette]] 95, March 2011, 102-107.</ref>

For prime ''p'', every integer-sided equiangular ''p''-gon is regular. Moreover, every integer-sided equiangular ''p''<sup>''k''</sup>-gon has ''p''-fold [[rotational symmetry]].<ref>McLean, K. Robin. "A powerful algebraic tool for equiangular polygons", ''Mathematical Gazette'' 88, November 2004, 513-514.</ref>

An ordered set of side lengths <math>(a_1, \dots , a_n)</math> gives rise to an equiangular ''n''-gon if and only if either of two equivalent conditions holds for the polynomial <math>a_1+a_2x+\cdots + a_{n-1}x^{n-2}+a_nx^{n-1}:</math> it equals zero at the complex value <math>e^{2\pi i/n};</math> it is divisible by <math>x^2-2x \cos (2\pi /n)+1.</math><ref>M. Bras-Amorós, M. Pujol: "Side Lengths of Equiangular Polygons (as seen by a coding theorist)", ''The American Mathematical Monthly'', vol. 122, n. 5, pp.&nbsp;476–478, May 2015. {{ISSN|0002-9890}}.</ref>
{{clear|left}}

== Direct equiangular polygons by sides ==
Direct equiangular polygons can be regular, isogonal, or lower symmetries. Examples for &lt;''p''/''q''&gt; are grouped into sections by ''p'' and subgrouped by density ''q''.

=== Equiangular triangles===
'''Equiangular triangles''' must be convex and have 60° internal angles. It is an [[equilateral triangle]] and a [[regular triangle]], {{angbr|3}}={3}. The only degree of freedom is edge-length.
<gallery>
Regular polygon 3 annotated.svg|Regular, {3}, r<sub>6</sub>
</gallery>

=== Equiangular quadrilaterals ===
[[File:Rectangle-2x3.svg|thumb|A rectangle dissected into a 2×3 array of squares<ref name="ball"/>]]
Direct '''equiangular quadrilaterals''' have 90° internal angles. The only equiangular quadrilaterals are [[rectangle]]s, {{angbr|4}}, and [[square]]s, {4}.

An equiangular quadrilateral with integer side lengths may be tiled by [[unit square]]s.<ref name="ball">{{citation|first=Derek|last=Ball|title=Equiangular polygons|journal=The Mathematical Gazette|volume=86|issue=507|year=2002|pages=396–407|doi=10.2307/3621131|jstor=3621131|s2cid=233358516}}.</ref>

<gallery>
Regular polygon 4 annotated.svg|Regular, {4}, r<sub>8</sub>
Spirolateral_2_90.svg|[[Spirolateral]] 2<sub>90°</sub>, p<sub>4</sub>
</gallery>

=== Equiangular pentagons ===
Direct '''equiangular pentagons''', {{angbr|5}} and {{angbr|5/2}}, have 108° and 36° internal angles respectively.

; 108° internal angle from an equiangular [[pentagon]], {{angbr|5}}
Equiangular pentagons can be [[regular pentagon|regular]], have bilateral symmetry, or no symmetry.
<gallery>
Equiangular pentagon 03.svg|Regular, r<sub>10</sub>
Equiangular pentagon 02.svg|Bilateral symmetry, i<sub>2</sub>
Equiangular pentagon 01.svg|No symmetry, a<sub>1</sub>
</gallery>

; 36° internal angles from an equiangular [[pentagram]], {{angbr|5/2}}
<gallery>
File:Regular star polygon 5-2.svg|Regular [[pentagram]], r<sub>10</sub>
Equiangular_pentagram1.svg|Irregular, d<sub>2</sub>
</gallery>

=== Equiangular hexagons ===
[[File:Equiangular_hexagon-1-2.svg|thumb|An equiangular hexagon with 1:2 edge length ratios, with equilateral triangles.<ref name="ball"/> This is [[spirolateral]] 2<sub>120°</sub>.]]
Direct '''equiangular hexagons''', {{angbr|6}} and {{angbr|6/2}}, have 120° and 60° internal angles respectively.

; 120° internal angles of an equiangular [[hexagon]], {{angbr|6}}:
An equiangular [[hexagon]] with integer side lengths may be tiled by unit [[equilateral triangle]]s.<ref name="ball"/>
<gallery>
Regular polygon 6 annotated.svg|Regular, {6}, r<sub>12</sub>
Spirolateral_2_120.svg|[[Spirolateral]] (1,2)<sub>120°</sub>, p<sub>6</sub>
Spirolateral_3_120.svg|Spirolateral (1…3)<sub>120°</sub>, g<sub>2</sub>
Spirolateral 1-2-2 120.svg|Spirolateral (1,2,2)<sub>120°</sub>, i<sub>4</sub>
Spirolateral 1-2-2-2-1-3 120.svg|Spirolateral (1,2,2,2,1,3)<sub>120°</sub>, p<sub>2</sub>
</gallery>

; 60° internal angles of an equiangular double-wound triangle, {{angbr|6/2}}:
<gallery>
Regular polygon 3 annotated.svg|Regular, degenerate, r<sub>6</sub>
Spirolateral 1-3 60.svg|[[Spirolateral]] (1,3)<sub>60°</sub>, p<sub>6</sub>
Spirolateral_2_60.svg|Spirolateral (1,2)<sub>60°</sub>, p<sub>6</sub>
Spirolateral 2-3 60.svg|Spirolateral (2,3)<sub>60°</sub>, p<sub>6</sub>
Spirolateral_1-2-3-4-3-2_60.svg|Spirolateral (1,2,3,4,3,2)<sub>60°</sub>, p<sub>2</sub>
</gallery>
{{clear|left}}

=== Equiangular heptagons ===
Direct '''equiangular heptagons''', {{angbr|7}}, {{angbr|7/2}}, and {{angbr|7/3}} have 128 4/7°, 77 1/7° and 25 5/7° internal angles respectively.

; 128.57° internal angles of an equiangular [[heptagon]], {{angbr|7}}:
<gallery>
Regular polygon 7 annotated.svg|Regular, {7}, r<sub>14</sub>
Equiangular heptagon.svg|Irregular, i<sub>2</sub>
</gallery>

; 77.14° internal angles of an equiangular [[heptagram]], {{angbr|7/2}}:
<gallery>
Regular star polygon 7-2.svg|Regular, r<sub>14</sub>
Equiangular heptagram1.svg|Irregular, i<sub>2</sub>
</gallery>

; 25.71° internal angles of an equiangular  [[heptagram]], {{angbr|7/3}}:
<gallery>
Regular star polygon 7-3.svg|Regular, r<sub>14</sub>
Equiangular heptagram2.svg|Irregular, i<sub>2</sub>
</gallery>

=== Equiangular octagons ===
Direct '''equiangular octagons''', {{angbr|8}}, {{angbr|8/2}} and {{angbr|8/3}}, have 135°, 90° and 45° internal angles respectively.

; 135° internal angles from an equiangular [[octagon]], {{angbr|8}}:
<gallery>
Regular polygon 8 annotated.svg|Regular, r<sub>16</sub>
Spirolateral_2_135.svg|[[Spirolateral]] (1,2)<sub>135°</sub>, p<sub>8</sub>
Spirolateral_4_135.svg|Spirolateral (1…4)<sub>135°</sub>, g<sub>2</sub>
Equiangular_octagon.svg|Unequal truncated square, p<sub>2</sub>
</gallery>

; 90° internal angles from an equiangular double-wound [[square]], {{angbr|8/2}}:
<gallery>
Regular polygon 4 annotated.svg|Regular degenerate, r<sub>8</sub>
Spirolateral 1-2-2-3-3-2-2-1 90.svg|Spirolateral (1,2,2,3,3,2,2,1)<sub>90°</sub>, d<sub>2</sub>
Spirolateral 2-1-3-2-2-3-1-2 90.svg|Spirolateral (2,1,3,2,2,3,1,2)<sub>90°</sub>, d<sub>2</sub>
</gallery>

; 45° internal angles from an equiangular [[octagram]], {{angbr|8/3}}:
<gallery>
Regular star polygon 8-3.svg|Regular, r<sub>16</sub>
Regular truncation 4 2.svg|Isogonal, p<sub>8</sub>
Regular truncation 4 -4.svg|Isogonal, p<sub>8</sub>
Spirolateral_2_45.svg|[[Spirolateral]], (1,2)<sub>45°</sub>, p<sub>8</sub>
Regular truncation 4 -0.2.svg|Isogonal, p<sub>8</sub>
Spirolateral_4_45.svg|Spirolateral (1…4)<sub>45°</sub>, g<sub>2</sub>
</gallery>

=== Equiangular enneagons ===
Direct '''equiangular enneagons''', {{angbr|9}}, {{angbr|9/2}}, {{angbr|9/3}}, and {{angbr|9/4}} have 140°, 100°, 60° and 20° internal angles respectively.

;140° internal angles from an equiangular enneagon {{angbr|9}}
<gallery>
Regular polygon 9 annotated.svg|Regular, r<sub>18</sub>
Spirolateral 1-1-3 140.svg|Spirolateral (1,1,3)<sub>140°</sub>, i<sub>6</sub>
</gallery>
;100° internal angles from an equiangular [[enneagram (geometry)|enneagram]], {{angbr|9/2}}:
<gallery>
Regular star polygon 9-2.svg|Regular {9/2}, p<sub>9</sub>
Spirolateral 1-1-5 140.svg|Spirolateral (1,1,5)<sub>100°</sub>, i<sub>6</sub>
File:Spirolateral 3 100.svg|Spirolateral 3<sub>100°</sub>, g<sub>3</sub>
</gallery>
;60° internal angles from an equiangular ''triple-wound triangle'', {{angbr|9/3}}:
<gallery>
Regular polygon 3 annotated.svg|Regular, degenerate, r<sub>6</sub>
Equiangular_triple-triangle1.svg|Irregular, a<sub>1</sub>
Equiangular_triple-triangle2.svg|Irregular, a<sub>1</sub>
Equiangular_triple-triangle3.svg|Irregular, a<sub>1</sub>
</gallery>

;20° internal angles from an equiangular [[enneagram (geometry)|enneagram]], {{angbr|9/4}}:
<gallery>
Regular star polygon 9-4.svg|Regular {9/4}, r<sub>18</sub>
Spirolateral 3 20.svg|[[Spirolateral]] 3<sub>20°</sub>, g<sub>3</sub>
Equiangular enneagram2.svg|Irregular, i<sub>2</sub>
</gallery>

=== Equiangular decagons ===
Direct '''equiangular decagons''', {{angbr|10}}, {{angbr|10/2}}, {{angbr|10/3}}, {{angbr|10/4}}, have 144°, 108°, 72° and 36° internal angles respectively.

;144° internal angles from an equiangular [[decagon]] {{angbr|10}}
<gallery>
Regular polygon 10 annotated.svg|Regular, r<sub>20</sub>
Spirolateral 2 144.svg|[[Spirolateral]] (1,2)<sub>144°</sub>, p<sub>10</sub>
Spirolateral_5_144.svg|Spirolateral (1…5)<sub>144°</sub>, g<sub>2</sub>
</gallery>

;108° internal angles from an equiangular double-wound [[pentagon]] {{angbr|10/2}}
<gallery>
Regular polygon 5 annotated.svg|Regular, degenerate
Spirolateral 2 108.svg|[[Spirolateral]] (1,2)<sub>108°</sub>, p<sub>10</sub>
Equiangular_double-pentagon1.svg|Irregular, p<sub>2</sub>
</gallery>

;72° internal angles from an equiangular [[Decagram (geometry)|decagram]] {{angbr|10/3}}
<gallery>
Regular star polygon 10-3.svg|Regular {10/3}, r<sub>20</sub>
Regular star truncation 5-3 3.svg|Isogonal, p<sub>10</sub>
Spirolateral_2_72.svg|[[Spirolateral]] (1,2)<sub>72°</sub>, p<sub>10</sub>
Equiangular_double-pentagon2.svg|Irregular, i<sub>4</sub>
Spirolateral 5 72.svg|Spirolateral (1…5)<sub>72°</sub>, g<sub>2</sub>
</gallery>

;36° internal angles from an equiangular double-wound [[pentagram]] {{angbr|10/4}}
<gallery>
Regular star polygon 5-2.svg|Regular, degenerate, r<sub>10</sub>
Spirolateral 2 36.svg|[[Spirolateral]] (1,2)<sub>36°</sub>, p<sub>10</sub>
Regular polygon truncation 5 3.svg|Isogonal, p<sub>10</sub>
Regular truncation 5 4.svg|Isogonal, p<sub>10</sub>
Equiangular double-pentagram3.svg|Irregular, p<sub>2</sub>
Equiangular_double-pentagon5.svg|Irregular, p<sub>2</sub>
Equiangular_double-pentagram2.svg|Irregular, p<sub>2</sub>
</gallery>

=== Equiangular hendecagons ===
Direct '''equiangular hendecagons''', {{angbr|11}}, {{angbr|11/2}}, {{angbr|11/3}}, {{angbr|11/4}}, and {{angbr|11/5}} have 147 3/11°, 114 6/11°, 81 9/11°, 49 1/11°, and 16 4/11° internal angles respectively.

;147° internal angles from an equiangular [[hendecagon]], {{angbr|11}}:
<gallery>
Regular polygon 11 annotated.svg|Regular, {11}, r<sub>22</sub>
</gallery>

;114° internal angles from an equiangular [[hendecagram]], {{angbr|11/2}}:
<gallery>
Regular star polygon 11-2.svg|Regular {11/2}, r<sub>22</sub>
</gallery>

;81° internal angles from an equiangular [[hendecagram]], {{angbr|11/3}}:
<gallery>
Regular star polygon 11-3.svg|Regular {11/3}, r<sub>22</sub>
</gallery>

;49° internal angles from an equiangular [[hendecagram]], {{angbr|11/4}}:
<gallery>
Regular star polygon 11-4.svg|Regular {11/4}, r<sub>22</sub>
</gallery>

;16° internal angles from an equiangular [[hendecagram]], {{angbr|11/5}}:
<gallery>
Regular star polygon 11-5.svg|Regular {11/5}, r<sub>22</sub>
</gallery>

=== Equiangular dodecagons ===
Direct '''equiangular dodecagons''', {{angbr|12}}, {{angbr|12/2}}, {{angbr|12/3}}, {{angbr|12/4}}, and {{angbr|12/5}} have 150°, 120°, 90°, 60°, and 30° internal angles respectively.

;150° internal angles from an equiangular [[dodecagon]], {{angbr|12}}:
Convex solutions with integer edge lengths may be tiled by [[pattern blocks]], squares, equilateral triangles, and 30° [[rhombus|rhombi]].<ref name="ball"/>
<gallery>
Regular polygon 12 annotated.svg|Regular, {12}, r<sub>24</sub>
Regular truncation 6 0.45.svg|Isogonal, p<sub>12</sub>
Spirolateral_2_150.svg|[[Spirolateral]] (1,2)<sub>150°</sub>, p<sub>12</sub> 
Spirolateral_3_150.svg|Spirolateral (1…3)<sub>150°</sub>, g<sub>4</sub> 
Spirolateral_4_150.svg|Spirolateral (1…4)<sub>150°</sub>, g<sub>3</sub> 
Spirolateral_6_150.svg|Spirolateral (1…6)<sub>150°</sub>, g<sub>2</sub> 
</gallery>

; 120° internal angles from an equiangular ''double-wound [[hexagon]]'', {{angbr|12/2}}
<gallery>
Regular polygon 6 annotated.svg|Regular degenerate, r<sub>12</sub>
Spirolateral 4 120.svg|[[Spirolateral]], (1…4)<sub>120°</sub>, g<sub>3</sub>
Equiangular double-hexagon3.svg|Irregular, d<sub>2</sub>
Equiangular double-hexagon2.svg|Irregular, d<sub>2</sub>
</gallery>

; 90° internal angles from an equiangular ''triple-wound [[square]]'', {{angbr|12/3}}
<gallery>
Regular polygon 4 annotated.svg|Regular, degenerate, r<sub>8</sub>
Spirolateral_3_90.svg|[[Spirolateral]] (1…3)<sub>90°</sub>, g<sub>2</sub> 
Spirolateral 2-3-4-90.svg|Spirolateral (2…4)<sub>90°</sub>, g<sub>4</sub>
Equiangular triple-square3.svg|Spirolateral (1,1,3)<sub>90°</sub>, i<sub>8</sub>
Spirolateral 1-2-2 90.svg|Spirolateral (1,2,2)<sub>90°</sub>, i<sub>8</sub>
Spirolateral_6_90.svg|Spirolateral (1…6)<sub>90°</sub>, g<sub>2</sub> 
Equiangular_triple-square1.svg|Irregular, a<sub>1</sub>
</gallery>

; 60° internal angles from an equiangular ''quadruple-wound [[triangle]]'', {{angbr|12/4}}
<gallery>
Regular polygon 3 annotated.svg|Regular, degenerate, r<sub>6</sub>
Equiangular double-hexagon4.svg|[[Spirolateral]] (1,3,5,1)<sub>60°</sub>, p<sub>6</sub>
Spirolateral 4 60.svg|Spirolateral (1…4)<sub>60°</sub>, g<sub>3</sub>
Equiangular_quadruple-triangle1.svg|Irregular, a<sub>1</sub>
</gallery>

; 30° internal angles from an equiangular ''[[dodecagram]]'', {{angbr|12/5}}
<gallery>
Regular star polygon 12-5.svg|Regular {12/5}, r<sub>24</sub>
Regular truncation 6 1.5.svg|Isogonal, p<sub>12</sub>
Spirolateral_2_30.svg|[[Spirolateral]] (1,2)<sub>30°</sub>, p<sub>12</sub>
Spirolateral_3_30.svg|Spirolateral (1…3)<sub>30°</sub>, g<sub>4</sub>
Spirolateral_4_30.svg|Spirolateral (1…4)<sub>30°</sub>, g<sub>3</sub>
Spirolateral_6_30.svg|Spirolateral (1…6)<sub>30°</sub>, g<sub>2</sub>
</gallery>

=== Equiangular tetradecagons ===
Direct '''equiangular tetradecagons''', {{angbr|14}}, {{angbr|14/2}}, {{angbr|14/3}}, {{angbr|14/4}}, and {{angbr|14/5}}, {{angbr|14/6}}, have 154 2/7°, 128 4/7°, 102 6/7°, 77 1/7°, 51 3/7° and 25 5/7° internal angles respectively.

;154.28° internal angles from an equiangular [[tetradecagon]], {{angbr|14}}:
<gallery>
Regular polygon 14 annotated.svg|Regular {14}, r<sub>28</sub>
File:Regular truncation 7 0.1.svg|Isogonal, t{7}, p<sub>14</sub>
</gallery>

;128.57° internal angles from an equiangular double-wound regular [[heptagon]], {{angbr|14/2}}:
<gallery>
Regular polygon 7 annotated.svg|Regular degenerate, r<sub>14</sub>
regular_star_truncation_7-5_3.svg|Isogonal, t{7/2}, p<sub>14</sub>
Spirolateral_2_129.svg|[[Spirolateral]] 2<sub>128.57°</sub>
</gallery>

;102.85° internal angles from an equiangular [[tetradecagram]], {{angbr|14/3}}:
<gallery>
Regular star polygon 14-3.svg|Regular {14/3}, r<sub>28</sub>
regular_star_truncation_7-3_3.svg|Isogonal t{7/3}, p<sub>14</sub>
</gallery>

;77.14° internal angles from an equiangular double-wound [[heptagram]] {{angbr|14/4}}:
<gallery>
Regular star polygon 7-2.svg|Regular degenerate, r<sub>14</sub>
regular_star_truncation_7-3_2.svg|Isogonal, p<sub>14</sub>
regular_star_truncation_7-3_4.svg|Isogonal, p<sub>14</sub>
File:Spirolateral 2 77.svg|[[Spirolateral]] 2<sub>77.14°</sub>
</gallery>

;51.43° internal angles from an equiangular [[tetradecagram]], {{angbr|14/5}}:
<gallery>
Regular star polygon 14-5.svg|Regular {14/5}, r<sub>28</sub>
regular_star_truncation_7-5_2.svg|Isogonal, p<sub>14</sub>
regular_star_truncation_7-5_4.svg|Isogonal, p<sub>14</sub>
</gallery>

;25.71° internal angles from an equiangular double-wound [[heptagram]], {{angbr|14/6}}:
<gallery>
Regular star polygon 7-3.svg|Regular degenerate, r<sub>14</sub>
Regular truncation 7 1000.svg|Isogonal, p<sub>14</sub>
File:Regular truncation 7 4.svg|Isogonal, p<sub>14</sub>
File:Regular truncation 7 -0.5.svg|Isogonal, p<sub>14</sub>
Equiangular_star-14-6.svg|Irregular, d<sub>2</sub>
</gallery>

=== Equiangular pentadecagons ===
Direct '''equiangular pentadecagons''', {{angbr|15}}, {{angbr|15/2}}, {{angbr|15/3}}, {{angbr|15/4}}, {{angbr|15/5}}, {{angbr|15/6}}, and {{angbr|15/7}}, have 156°, 132°, 108°, 84°, 60° and 12° internal angles respectively.

;156° internal angles from an equiangular pentadecagon, {{angbr|15}}:
<gallery>
Regular polygon 15 annotated.svg|Regular, {15}, r<sub>30</sub>
</gallery>

;132° internal angles from an equiangular [[pentadecagram]], {{angbr|15/2}}:
<gallery>
Regular star polygon 15-2.svg|Regular, {15/2}, r<sub>30</sub>
</gallery>

;108° internal angles from an equiangular triple-wound pentagon, {{angbr|15/3}}:
<gallery>
Regular polygon 5 annotated.svg|Regular, degenerate, r<sub>10</sub>
Equiangular_triple-pentagon1.svg|[[spirolateral]] (1…3)<sub>108°</sub>, g<sub>5</sub>
</gallery>

;84° internal angles from an equiangular pentadecagram, {{angbr|15/4}}:
<gallery>
Regular star polygon 15-4.svg|Regular, {15/4}, r<sub>30</sub>
</gallery>

;60° internal angles from an equiangular 5-wound [[triangle]], {{angbr|15/5}}:
<gallery>
Regular polygon 3 annotated.svg|Regular, degenerate, r<sub>6</sub>
Equiangular 5-wound triangle1.svg|Irregular, a<sub>1</sub>
</gallery>

;36° internal angles from an equiangular triple-wound [[pentagram]], {{angbr|15/6}}:
<gallery>
Regular star polygon 5-2.svg|Regular, degenerate, r<sub>10</sub>
Equiangular triple-pentagram2.svg|Irregular, a<sub>1</sub>
File:Spirolateral 4 36.svg|Spirolateral (1…4)<sub>36°</sub>, g<sub>5</sub>
</gallery>

;12° internal angles from an equiangular pentadecagram, {{angbr|15/7}}:
<gallery>
Regular star polygon 15-7.svg|Regular, {15/7}, r<sub>30</sub>
</gallery>

=== Equiangular hexadecagons ===
Direct '''equiangular hexadecagons''', {{angbr|16}}, {{angbr|16/2}}, {{angbr|16/3}}, {{angbr|16/4}}, {{angbr|16/5}}, {{angbr|16/6}}, and {{angbr|16/7}}, have 157.5°, 135°, 112.5°, 90°, 67.5° 45° and 22.5° internal angles respectively.

;157.5° internal angles from an equiangular [[hexadecagon]], {{angbr|16}}:
<gallery>
Regular polygon 16 annotated.svg|Regular, {16}, r<sub>32</sub>
File:Regular truncation 8 0.45.svg|Isogonal, t{8}, p<sub>16</sub>
File:Spirolateral 4 1575.svg|[[Spirolateral]] (1…4)<sub>157.5°</sub>, g<sub>4</sub>
</gallery>

;135° internal angles from an equiangular double-wound octagon, {{angbr|16/2}}:
<gallery>
Regular polygon 8 annotated.svg|Regular, degenerate, r<sub>16</sub>
Equiangular double octagon1.svg|Irregular, p<sub>16</sub>
</gallery>

;112.5° internal angles from an equiangular [[hexadecagram]], {{angbr|16/3}}:
<gallery>
Regular star polygon 16-3.svg|Regular, {16/3}, r<sub>32</sub>
</gallery>

;90° internal angles from an equiangular 4-wound square, {{angbr|16/4}}:
<gallery>
Regular polygon 4 annotated.svg|Regular, degenerate, r<sub>8</sub>
Equiangular 4-wound square1.svg|Irregular, a<sub>1</sub>
</gallery>

;67.5° internal angles from an equiangular [[hexadecagram]], {{angbr|16/5}}:
<gallery>
Regular star polygon 16-5.svg|Regular, {16/5}, r<sub>32</sub>
</gallery>

;45° internal angles from an equiangular double-wound regular [[octagram]], {{angbr|16/6}}:
<gallery>
Regular star polygon 8-3.svg|Regular, degenerate, r<sub>16</sub>
Spirolateral 3 45.svg|[[spirolateral]] (1…3)<sub>45°</sub>, g<sub>8</sub>
</gallery>

;22.5° internal angles from an equiangular [[hexadecagram]], {{angbr|16/7}}:
<gallery>
Regular star polygon 16-7.svg|Regular, {16/7}, r<sub>32</sub>
File:Regular truncation 8 -10.svg|Isogonal, p<sub>16</sub>
</gallery>

=== Equiangular octadecagons ===
Direct '''equiangular octadecagons''', <18}, {{angbr|18/2}}, {{angbr|18/3}}, {{angbr|18/4}}, {{angbr|18/5}}, {{angbr|18/6}}, {{angbr|18/7}}, and {{angbr|18/8}}, have 160°, 140°, 120°, 100°, 80°, 60°, 40° and 20° internal angles respectively.

;160° internal angles from an equiangular [[octadecagon]], {{angbr|18}}:
<gallery>
Regular polygon 18 annotated.svg|Regular, {18}, r<sub>36</sub>
File:Regular truncation 9 0.1.svg|Isogonal, t{9}, p<sub>18</sub>
</gallery>

;140° internal angles from an equiangular double-wound [[enneagon]], {{angbr|18/2}}:
<gallery>
Regular polygon 9.svg|Regular, degenerate
Spirolateral 2 140.svg|[[Spirolateral]] 2<sub>140°</sub>, p<sub>18</sub>
</gallery>

; 120° internal angles of an equiangular 3-wound hexagon {{angbr|18/3}}:
<gallery>
Regular polygon 6.svg|Regular, degenerate, r<sub>18</sub>
Equilateral triple-wound-hexagon1.svg|irregular, a<sub>1</sub>
</gallery>

; 100° internal angles of an equiangular double-wound [[Enneagram (geometry)|enneagram]] {{angbr|18/4}}:
<gallery>
Regular star polygon 9-2.svg|Regular, degenerate, r<sub>18</sub>
File:Spirolateral_6_100.svg|[[Spirolateral]] 2<sub>100°</sub>, g<sub>3</sub>
</gallery>

; 80° internal angles of an equiangular [[octadecagram]] {18/5}:
<gallery>
Regular star polygon 18-5.svg|Regular, {18/5}, r<sub>36</sub>
</gallery>

; 60° internal angles of an equiangular 6-wound triangle {{angbr|18/6}}:
<gallery>
Regular polygon 3.svg|Regular, degenerate, r<sub>6</sub>
Equilateral 6-wound-triangle1.svg|irregular, a<sub>1</sub>
</gallery>

; 40° internal angles of an equiangular [[octadecagram]] {{angbr|18/7}}:
<gallery>
Regular star polygon 18-7.svg|Regular, {18/7}, r<sub>36</sub>
regular_star_truncation_9-7_2.svg|Isogonal, p<sub>18</sub>
regular_star_truncation_9-7_4.svg|Isogonal, p<sub>18</sub>
regular_star_truncation_9-7_5.svg|Isogonal, p<sub>18</sub>
</gallery>

; 20° internal angles of an equiangular double-wound [[Enneagram (geometry)|enneagram]] {{angbr|18/8}}:
<gallery>
Regular star polygon 9-4.svg|Regular, degenerate, r<sub>18</sub>
regular_polygon_truncation_9_5.svg|Isogonal, p<sub>18</sub>
regular_polygon_truncation_9_4.svg|Isogonal, p<sub>18</sub>
regular_polygon_truncation_9_3.svg|Isogonal, p<sub>18</sub>
regular_polygon_truncation_9_2.svg|Isogonal, p<sub>18</sub>
Spirolateral 2 20.svg|[[Spirolateral]] 2<sub>20°</sub>, p<sub>18</sub>
File:Spirolateral 6 20.svg|Spirolateral 6<sub>20°</sub>, g<sub>3</sub>
</gallery>

=== Equiangular icosagons ===
Direct '''equiangular icosagon''', {{angbr|20}}, {{angbr|20/3}}, {{angbr|20/4}}, {{angbr|20/5}}, {{angbr|20/6}}, {{angbr|20/7}}, and {{angbr|20/9}}, have 162°, 126°, 108°, 90°, 72°, 54° and 18° internal angles respectively.
;162° internal angles from an equiangular [[icosagon]], {{angbr|20}}:
<gallery>
Regular polygon 20 annotated.svg|Regular, {20}, r<sub>40</sub>
Spirolateral 1-3 162.svg|[[Spirolateral]] (1,3)<sub>162°</sub>, p<sub>20</sub>
</gallery>

;144° internal angles from an equiangular double-wound [[decagon]], {{angbr|20/2}}:
<gallery>
Regular polygon 10 annotated.svg|Regular, degenerate, r<sub>20</sub>
Spirolateral_4_144.svg|[[Spirolateral]] (1…4)<sub>144°</sub>, g<sub>5</sub>
</gallery>

;126° internal angles from an equiangular [[icosagram]], {{angbr|20/3}}:
<gallery>
Regular star polygon 20-3.svg|Regular {20/3}, p<sub>40</sub>
Spirolateral_1-3_126.svg|[[Spirolateral]] (1,3)<sub>126°</sub>, p<sub>20</sub>
</gallery>

;108° internal angles from an equiangular 4-wound [[pentagon]], {{angbr|20/4}}:
<gallery>
Regular polygon 5 annotated.svg|Regular degenerate, r<sub>10</sub>
Spirolateral 4 108.svg|[[Spirolateral]] (1…4)<sub>108°</sub>, g<sub>5</sub>
Equiangular quadruple-pentagon1.svg|Irregular, a<sub>1</sub>
</gallery>

;90° internal angles from an equiangular 5-wound [[square]], {{angbr|20/5}}:
<gallery>
Regular polygon 4 annotated.svg|Regular degenerate, r<sub>8</sub>
Spirolateral 5 90.svg|[[Spirolateral]] (1…5)<sub>90°</sub>, g<sub>4</sub>
Spirolateral 1-2-3-2-1 90.svg|[[Spirolateral]] (1,2,3,2,1)<sub>90°</sub>, i<sub>8</sub>
</gallery>

;72° internal angles from an equiangular double-wound [[Decagram (geometry)|decagram]], {{angbr|20/6}}:
<gallery>
Regular star polygon 10-3.svg|Regular degenerate, r<sub>20</sub>
Spirolateral 2 72.svg|[[Spirolateral]] (1,2)<sub>72°</sub>, p<sub>10</sub>
Spirolateral 4 72.svg|Spirolateral (1…4)<sub>72°</sub>, g<sub>5</sub>
</gallery>

;54° internal angles from an equiangular [[icosagram]], {{angbr|20/7}}:
<gallery>
Regular star polygon 20-7.svg|Regular {20/7}, r<sub>40</sub>
regular_star_truncation_10-3_2.svg|Isogonal, p<sub>20</sub>
regular_star_truncation_10-3_3.svg|Isogonal, p<sub>20</sub>
regular_star_truncation_10-3_5.svg|Isogonal, p<sub>20</sub>
</gallery>

;36° internal angles from an equiangular quadruple-wound [[pentagram]], {{angbr|20/8}}:
<gallery>
Regular star polygon 5-2.svg|Regular degenerate, r<sub>10</sub>
Spirolateral_4_36.svg|[[Spirolateral]] (1…4)<sub>36°</sub>, g<sub>5</sub>
Equiangular quadruple-pentagram1.svg|irregular, a<sub>1</sub>
</gallery>

;18° internal angles from an equiangular [[icosagram]], {{angbr|20/9}}:
<gallery>
Regular star polygon 20-9.svg|Regular {20/9}, r<sub>40</sub>
regular_polygon_truncation_10_5.svg|Isogonal, p<sub>20</sub>
regular_polygon_truncation_10_4.svg|Isogonal, p<sub>20</sub>
regular_polygon_truncation_10_3.svg|Isogonal, p<sub>20</sub>
regular_polygon_truncation_10_2.svg|Isogonal, p<sub>20</sub>
</gallery>

== See also ==
* [[Spirolateral]]

==References==
{{reflist}}
*Williams, R. ''The Geometrical Foundation of Natural Structure: A Source Book of Design''. New York: [[Dover Publications]], 1979. p.&nbsp;32

==External links==
*[http://www.cut-the-knot.org/Curriculum/Geometry/EquiangularPoly.shtml A Property of Equiangular Polygons: What Is It About?] a discussion of Viviani's theorem at [[Cut-the-knot]].
* {{MathWorld|EquiangularPolygon|Equiangular Polygon}}

{{polygons}}

[[Category:Types of polygons]]