Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, an equiangular polygon is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also

equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

) then it is a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

Isogonal polygon In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

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 turns. Convex equiangular polygons are always direct. An indirect equiangular polygon can include angles turning right or left in any combination. A

skew Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not ...

equiangular polygon may be isogonal, but can't be considered direct since it is nonplanar. A

spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...

''n''_θ 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 θ.

Construction

An ''equiangular polygon'' can be constructed from a

regular star polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

where edges are extended as infinite lines. 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. 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 θ°, moving alternate edges can invert all vertices into supplementary angles, 180-θ°. 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.

Equiangular polygon theorem

For a ''convex equiangular'' ''p''-gon, each

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

is 180(1-2/''p'')°; this is the ''equiangular polygon theorem''. For a direct equiangular ''p''/''q''

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

''q'', each internal angle is 180(1-2''q''/''p'')°, with 1<2''q''<''p''. For ''w''=gcd(''p'',''q'')>1, this represents a ''w''-wound (''p''/''w'')/(''q''/''w'') star polygon, which is degenerate for the regular case. A concave ''indirect equiangular'' (''p''_''r''+''p''_''l'')-gon, with ''p''_''r'' right turn vertices and ''p''_''l'' left turn vertices, will have internal angles of 180(1-2/, ''p''_''r''-''p''_''l'', ))°, regardless of their sequence. An ''indirect star equiangular'' (''p''_''r''+''p''_''l'')-gon, with ''p''_''r'' right turn vertices and ''p''_''l'' left turn vertices and ''q'' total turns, will have internal angles of 180(1-2''q''/, ''p''_''r''-''p''_''l'', ))°, 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 <''p''> or <''p''/''q''>, like

s and

s , containing ''p'' vertices, and stars having

''q''. Convex equiangular ''p''-gons <''p''> have internal angles 180(1-2/''p'')°, while direct star equiangular polygons, <''p''/''q''>, have internal angles 180(1-2''q''/''p'')°. A concave indirect equiangular ''p''-gon can be given the notation <''p''-2''c''>, with ''c'' counter-turn vertices. For example, <6-2> is a hexagon with 90° internal angles of the difference, <4>, 1 counter-turned vertex. A multiturn indirect equilateral ''p''-gon can be given the notation <^''p''-2''c''/_''q''> with ''c'' counter turn vertices, and ''q'' total turns. An equiangular polygon <''p''-''p''> is a ''p''-gon with undefined internal angles θ, but can be expressed explicitly as <''p''-''p''>_θ.

Other properties

Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...

holds for equiangular polygons:Elias Abboud "On Viviani’s Theorem and its Extensions"
pp. 2, 11 :''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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

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. For prime ''p'', every integer-sided equiangular ''p''-gon is regular. Moreover, every integer-sided equiangular ''p''^''k''-gon has ''p''-fold

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

. An ordered set of side lengths

(a_1, \dots , a_n)

gives rise to an equiangular ''n''-gon if and only if either of two equivalent conditions holds for the polynomial

a_1+a_2x+\cdots + a_x^+a_nx^:

it equals zero at the complex value

e^;

it is divisible by

x^2-2x \cos (2\pi /n)+1.

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. 476–478, May 2015. .

Direct equiangular polygons by sides

Direct equiangular polygons can be regular, isogonal, or lower symmetries. Examples for <''p''/''q''> 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 In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

and a

regular triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each ...

, <3>=. The only degree of freedom is edge-length. Regular polygon 3 annotated.svg, Regular, , r₆

Equiangular quadrilaterals

Direct equiangular quadrilaterals have 90° internal angles. The only equiangular quadrilaterals are

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

s, <4>, and

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

s, . An equiangular quadrilateral with integer side lengths may be tiled by

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinate ...

s.. Regular polygon 4 annotated.svg, Regular, , r₈ Spirolateral_2_90.svg,

Spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...

2_90°, p₄

Equiangular pentagons

Direct equiangular pentagons, <5> and <5/2>, have 108° and 36° internal angles respectively. ; 108° internal angle from an equiangular

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

, <5> Equiangular pentagons can can be regular, have bilateral symmetry, or no symmetry. Equiangular pentagon 03.svg, Regular, r₁₀ Equiangular pentagon 02.svg, Bilateral symmetry, i₂ Equiangular pentagon 01.svg, No symmetry, a₁ ; 36° internal angles from an equiangular

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

, <5/2> File:Regular star polygon 5-2.svg, Regular

, r₁₀ Equiangular_pentagram1.svg, Irregular, d₂

Equiangular hexagons

Direct equiangular hexagons, <6> and <6/2>, have 120° and 60° internal angles respectively. ; 120° internal angles of an equiangular

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

, <6>: An equiangular

with integer side lengths may be tiled by unit

s. Regular polygon 6 annotated.svg, Regular, , r₁₂ Spirolateral_2_120.svg,

(1,2)_120°, p₆ Spirolateral_3_120.svg, Spirolateral (1…3)_120°, g₂ Spirolateral 1-2-2 120.svg, Spirolateral (1,2,2)_120°, i₄ Spirolateral 1-2-2-2-1-3 120.svg, Spirolateral (1,2,2,2,1,3)_120°, p₂ ; 60° internal angles of an equiangular double-wound triangle, <6/2>: Regular polygon 3 annotated.svg, Regular, degenerate, r₆ Spirolateral 1-3 60.svg,

(1,3)_60°, p₆ Spirolateral_2_60.svg, Spirolateral (1,2)_60°, p₆ Spirolateral 2-3 60.svg, Spirolateral (2,3)_60°, p₆ Spirolateral_1-2-3-4-3-2_60.svg, Spirolateral (1,2,3,4,3,2)_60°, p₂

Equiangular heptagons

Direct equiangular heptagons, <7>, <7/2>, and <7/3> have 128 4/7°, 77 1/7° and 25 5/7° internal angles respectively. ; 128.57° internal angles of an equiangular

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...

, <7>: Regular polygon 7 annotated.svg, Regular, , r₁₄ Equiangular heptagon.svg, Irregular, i₂ ; 77.14° internal angles of an equiangular

heptagram A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ� ...

, <7/2>: Regular star polygon 7-2.svg, Regular, r₁₄ Equiangular heptagram1.svg, Irregular, i₂ ; 25.71° internal angles of an equiangular

, <7/3>: Regular star polygon 7-3.svg, Regular, r₁₄ Equiangular heptagram2.svg, Irregular, i₂

Equiangular octagons

Direct equiangular octagons, <8>, <8/2> and <8/3>, have 135°, 90° and 45° internal angles respectively. ; 135° internal angles from an equiangular

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

, <8>: Regular polygon 8 annotated.svg, Regular, r₁₆ Spirolateral_2_135.svg,

(1,2)_135°, p₈ Spirolateral_4_135.svg, Spirolateral (1…4)_135°, g₂ Equiangular_octagon.svg, Unequal truncated square, p₂ ; 90° internal angles from an equiangular double-wound

, <8/2>: Regular polygon 4 annotated.svg, Regular degenerate, r₈ Spirolateral 1-2-2-3-3-2-2-1 90.svg, Spirolateral (1,2,2,3,3,2,2,1)_90°, d₂ Spirolateral 2-1-3-2-2-3-1-2 90.svg, Spirolateral (2,1,3,2,2,3,1,2)_90°, d₂ ; 45° internal angles from an equiangular

octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Deta ...

, <8/3>: Regular star polygon 8-3.svg, Regular, r₁₆ Regular truncation 4 2.svg, Isogonal, p₈ Regular truncation 4 -4.svg, Isogonal, p₈ Spirolateral_2_45.svg,

, (1,2)_45°, p₈ Regular truncation 4 -0.2.svg, Isogonal, p₈ Spirolateral_4_45.svg, Spirolateral (1…4)_45°, g₂

Equiangular enneagons

Direct equiangular enneagons, <9>, <9/2>, <9/3>, and <9/4> have 140°, 100°, 60° and 20° internal angles respectively. ;140° internal angles from an equiangular enneagon <9> Regular polygon 9 annotated.svg, Regular, r₁₈ Spirolateral 1-1-3 140.svg, Spirolateral (1,1,3)_140°, i₆ ;100° internal angles from an equiangular enneagram, <9/2>: Regular star polygon 9-2.svg, Regular , p₉ Spirolateral 1-1-5 140.svg, Spirolateral (1,1,5)_100°, i₆ File:Spirolateral 3 100.svg, Spirolateral 3_100°, g₃ ;60° internal angles from an equiangular ''triple-wound triangle'', <9/3>: Regular polygon 3 annotated.svg, Regular, degenerate, r₆ Equiangular_triple-triangle1.svg, Irregular, a₁ Equiangular_triple-triangle2.svg, Irregular, a₁ Equiangular_triple-triangle3.svg, Irregular, a₁ ;20° internal angles from an equiangular enneagram, <9/4>: Regular star polygon 9-4.svg, Regular , r₁₈ Spirolateral 3 20.svg,

3_20°, g₃ Equiangular enneagram2.svg, Irregular, i₂

Equiangular decagons

Direct equiangular decagons, <10>, <10/2>, <10/3>, <10/4>, have 144°, 108°, 72° and 36° internal angles respectively. ;144° internal angles from an equiangular

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

<10> Regular polygon 10 annotated.svg, Regular, r₂₀ Spirolateral 2 144.svg,

(1,2)_144°, p₁₀ Spirolateral_5_144.svg, Spirolateral (1…5)_144°, g₂ ;108° internal angles from an equiangular double-wound

<10/2> Regular polygon 5 annotated.svg, Regular, degenerate Spirolateral 2 108.svg,

(1,2)_108°, p₁₀ Equiangular_double-pentagon1.svg, Irregular, p₂ ;72° internal angles from an equiangular decagram <10/3> Regular star polygon 10-3.svg, Regular , r₂₀ Regular star truncation 5-3 3.svg, Isogonal, p₁₀ Spirolateral_2_72.svg,

(1,2)_72°, p₁₀ Equiangular_double-pentagon2.svg, Irregular, i₄ Spirolateral 5 72.svg, Spirolateral (1…5)_72°, g₂ ;36° internal angles from an equiangular double-wound

<10/4> Regular star polygon 5-2.svg, Regular, degenerate, r₁₀ Spirolateral 2 36.svg,

(1,2)_36°, p₁₀ Regular polygon truncation 5 3.svg, Isogonal, p₁₀ Regular truncation 5 4.svg, Isogonal, p₁₀ Equiangular double-pentagram3.svg, Irregular, p₂ Equiangular_double-pentagon5.svg, Irregular, p₂ Equiangular_double-pentagram2.svg, Irregular, p₂

Equiangular hendecagons

Direct equiangular hendecagons, <11>, <11/2>, <11/3>, <11/4>, and <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 In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ''hendecagon'', from Greek ''hendeka'' "eleven" and ''–gon'' "corner", is often preferred to the hybrid ''undecagon'', whose first part is f ...

, <11>: Regular polygon 11 annotated.svg, Regular, , r₂₂ ;114° internal angles from an equiangular

hendecagram In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices. The name ''hendecagram'' combines a Greek numeral prefix, '' hendeca-'', with the Greek suffix ''-gram''. The ''hendeca-'' prefix derives fro ...

, <11/2>: Regular star polygon 11-2.svg, Regular , r₂₂ ;81° internal angles from an equiangular

, <11/3>: Regular star polygon 11-3.svg, Regular , r₂₂ ;49° internal angles from an equiangular

, <11/4>: Regular star polygon 11-4.svg, Regular , r₂₂ ;16° internal angles from an equiangular

, <11/5>: Regular star polygon 11-5.svg, Regular , r₂₂

Equiangular dodecagons

Direct equiangular dodecagons, <12>, <12/2>, <12/3>, <12/4>, and <12/5> have 150°, 120°, 90°, 60°, and 30° internal angles respectively. ;150° internal angles from an equiangular

dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...

, <12>: Convex solutions with integer edge lengths may be tiled by

pattern blocks Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of m ...

, squares, equilateral triangles, and 30°

rhombi In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

. Regular polygon 12 annotated.svg, Regular, , r₂₄ Regular truncation 6 0.45.svg, Isogonal, p₁₂ Spirolateral_2_150.svg,

(1,2)_150°, p₁₂ Spirolateral_3_150.svg, Spirolateral (1…3)_150°, g₄ Spirolateral_4_150.svg, Spirolateral (1…4)_150°, g₃ Spirolateral_6_150.svg, Spirolateral (1…6)_150°, g₂ ; 120° internal angles from an equiangular ''double-wound

'', <12/2> Regular polygon 6 annotated.svg, Regular degenerate, r₁₂ Spirolateral 4 120.svg,

, (1…4)_120°, g₃ Equiangular double-hexagon3.svg, Irregular, d₂ Equiangular double-hexagon2.svg, Irregular, d₂ ; 90° internal angles from an equiangular ''triple-wound

'', <12/3> Regular polygon 4 annotated.svg, Regular, degenerate, r₈ Spirolateral_3_90.svg,

(1…3)_90°, g₂ Spirolateral 2-3-4-90.svg, Spirolateral (2…4)_90°, g₄ Equiangular triple-square3.svg, Spirolateral (1,1,3)_90°, i₈ Spirolateral 1-2-2 90.svg, Spirolateral (1,2,2)_90°, i₈ Spirolateral_6_90.svg, Spirolateral (1…6)_90°, g₂ Equiangular_triple-square1.svg, Irregular, a₁ ; 60° internal angles from an equiangular ''quadruple-wound

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

'', <12/4> Regular polygon 3 annotated.svg, Regular, degenerate, r₆ Equiangular double-hexagon4.svg,

(1,3,5,1)_60°, p₆ Spirolateral 4 60.svg, Spirolateral (1…4)_60°, g₃ Equiangular_quadruple-triangle1.svg, Irregular, a₁ ; 30° internal angles from an equiangular ''

dodecagram In geometry, a dodecagram (γραμμή
Henry George Liddell, Robe ...

'', <12/5> Regular star polygon 12-5.svg, Regular , r₂₄ Regular truncation 6 1.5.svg, Isogonal, p₁₂ Spirolateral_2_30.svg,

(1,2)_30°, p₁₂ Spirolateral_3_30.svg, Spirolateral (1…3)_30°, g₄ Spirolateral_4_30.svg, Spirolateral (1…4)_30°, g₃ Spirolateral_6_30.svg, Spirolateral (1…6)_30°, g₂

Equiangular tetradecagons

Direct equiangular tetradecagons, <14>, <14/2>, <14/3>, <14/4>, and <14/5>, <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 In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. Regular tetradecagon A '' regular tetradecagon'' has Schläfli symbol and can be constructed as a quasiregular truncated heptagon, t, which alternates two t ...

, <14>: Regular polygon 14 annotated.svg, Regular , r₂₈ File:Regular truncation 7 0.1.svg, Isogonal, t, p₁₄ ;128.57° internal angles from an equiangular double-wound regular

, <14/2>: Regular polygon 7 annotated.svg, Regular degenerate, r₁₄ regular_star_truncation_7-5_3.svg, Isogonal, t, p₁₄ Spirolateral_2_129.svg,

2_128.57° ;102.85° internal angles from an equiangular tetradecagram, <14/3>: Regular star polygon 14-3.svg, Regular , r₂₈ regular_star_truncation_7-3_3.svg, Isogonal t, p₁₄ ;77.14° internal angles from an equiangular double-wound

<14/4>: Regular star polygon 7-2.svg, Regular degenerate, r₁₄ regular_star_truncation_7-3_2.svg, Isogonal, p₁₄ regular_star_truncation_7-3_4.svg, Isogonal, p₁₄ File:Spirolateral 2 77.svg,

2_77.14° ;51.43° internal angles from an equiangular tetradecagram, <14/5>: Regular star polygon 14-5.svg, Regular , r₂₈ regular_star_truncation_7-5_2.svg, Isogonal, p₁₄ regular_star_truncation_7-5_4.svg, Isogonal, p₁₄ ;25.71° internal angles from an equiangular double-wound

, <14/6>: Regular star polygon 7-3.svg, Regular degenerate, r₁₄ Regular truncation 7 1000.svg, Isogonal, p₁₄ File:Regular truncation 7 4.svg, Isogonal, p₁₄ File:Regular truncation 7 -0.5.svg, Isogonal, p₁₄ Equiangular_star-14-6.svg, Irregular, d₂

Equiangular pentadecagons

Direct equiangular pentadecagons, <15>, <15/2>, <15/3>, <15/4>, <15/5>, <15/6>, and <15/7>, have 156°, 132°, 108°, 84°, 60° and 12° internal angles respectively. ;156° internal angles from an equiangular pentadecagon, <15>: Regular polygon 15 annotated.svg, Regular, , r₃₀ ;132° internal angles from an equiangular pentadecagram, <15/2>: Regular star polygon 15-2.svg, Regular, , r₃₀ ;108° internal angles from an equiangular triple-wound pentagon, <15/3>: Regular polygon 5 annotated.svg, Regular, degenerate, r₁₀ Equiangular_triple-pentagon1.svg,

(1…3)_108°, g₅ ;84° internal angles from an equiangular pentadecagram, <15/4>: Regular star polygon 15-4.svg, Regular, , r₃₀ ;60° internal angles from an equiangular 5-wound

, <15/5>: Regular polygon 3 annotated.svg, Regular, degenerate, r₆ Equiangular 5-wound triangle1.svg, Irregular, a₁ ;36° internal angles from an equiangular triple-wound

, <15/6>: Regular star polygon 5-2.svg, Regular, degenerate, r₁₀ Equiangular triple-pentagram2.svg, Irregular, a₁ File:Spirolateral 4 36.svg, Spirolateral (1…4)_36°, g₅ ;12° internal angles from an equiangular pentadecagram, <15/7>: Regular star polygon 15-7.svg, Regular, , r₃₀

Equiangular hexadecagons

Direct equiangular hexadecagons, <16>, <16/2>, <16/3>, <16/4>, <16/5>, <16/6>, and <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, <16>: Regular polygon 16 annotated.svg, Regular, , r₃₂ File:Regular truncation 8 0.45.svg, Isogonal, t, p₁₆ File:Spirolateral 4 1575.svg,

(1…4)_157.5°, g₄ ;135° internal angles from an equiangular double-wound octagon, <16/2>: Regular polygon 8 annotated.svg, Regular, degenerate, r₁₆ Equiangular double octagon1.svg, Irregular, p₁₆ ;112.5° internal angles from an equiangular

hexadecagram In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. Regular hexadecagon A '' regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli sym ...

, <16/3>: Regular star polygon 16-3.svg, Regular, , r₃₂ ;90° internal angles from an equiangular 4-wound square, <16/4>: Regular polygon 4 annotated.svg, Regular, degenerate, r₈ Equiangular 4-wound square1.svg, Irregular, a₁ ;67.5° internal angles from an equiangular

, <16/5>: Regular star polygon 16-5.svg, Regular, , r₃₂ ;45° internal angles from an equiangular double-wound regular

, <16/6>: Regular star polygon 8-3.svg, Regular, degenerate, r₁₆ Spirolateral 3 45.svg,

(1…3)_45°, g₈ ;22.5° internal angles from an equiangular

, <16/7>: Regular star polygon 16-7.svg, Regular, , r₃₂ File:Regular truncation 8 -10.svg, Isogonal, p₁₆

Equiangular octadecagons

Direct equiangular octadecagons, <18}, <18/2>, <18/3>, <18/4>, <18/5>, <18/6>, <18/7>, and <18/8>, have 160°, 140°, 120°, 100°, 80°, 60°, 40° and 20° internal angles respectively. ;160° internal angles from an equiangular

octadecagon In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates tw ...

, <18>: Regular polygon 18 annotated.svg, Regular, , r₃₆ File:Regular truncation 9 0.1.svg, Isogonal, t, p₁₈ ;140° internal angles from an equiangular double-wound

enneagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...

, <18/2>: Regular polygon 9.svg, Regular, degenerate Spirolateral 2 140.svg,

2_140°, p₁₈ ; 120° internal angles of an equiangular 3-wound hexagon <18/3>: Regular polygon 6.svg, Regular, degenerate, r₁₈ Equilateral triple-wound-hexagon1.svg, irregular, a₁ ; 100° internal angles of an equiangular double-wound enneagram <18/4>: Regular star polygon 9-2.svg, Regular, degenerate, r₁₈ File:Spirolateral_6_100.svg,

2_100°, g₃ ; 80° internal angles of an equiangular octadecagram : Regular star polygon 18-5.svg, Regular, , r₃₆ ; 60° internal angles of an equiangular 6-wound triangle <18/6>: Regular polygon 3.svg, Regular, degenerate, r₆ Equilateral 6-wound-triangle1.svg, irregular, a₁ ; 40° internal angles of an equiangular octadecagram <18/7>: Regular star polygon 18-7.svg, Regular, , r₃₆ regular_star_truncation_9-7_2.svg, Isogonal, p₁₈ regular_star_truncation_9-7_4.svg, Isogonal, p₁₈ regular_star_truncation_9-7_5.svg, Isogonal, p₁₈ ; 20° internal angles of an equiangular double-wound enneagram <18/8>: Regular star polygon 9-4.svg, Regular, degenerate, r₁₈ regular_polygon_truncation_9_5.svg, Isogonal, p₁₈ regular_polygon_truncation_9_4.svg, Isogonal, p₁₈ regular_polygon_truncation_9_3.svg, Isogonal, p₁₈ regular_polygon_truncation_9_2.svg, Isogonal, p₁₈ Spirolateral 2 20.svg,

2_20°, p₁₈ File:Spirolateral 6 20.svg, Spirolateral 6_20°, g₃

Equiangular icosagons

Direct equiangular icosagon, <20>, <20/3>, <20/4>, <20/5>, <20/6>, <20/7>, and <20/9>, have 162°, 126°, 108°, 90°, 72°, 54° and 18° internal angles respectively. ;162° internal angles from an equiangular icosagon, <20>: Regular polygon 20 annotated.svg, Regular, , r₄₀ Spirolateral 1-3 162.svg,

(1,3)_162°, p₂₀ ;144° internal angles from an equiangular double-wound

, <20/2>: Regular polygon 10 annotated.svg, Regular, degenerate, r₂₀ Spirolateral_4_144.svg,

(1…4)_144°, g₅ ;126° internal angles from an equiangular

icosagram In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. Regular icosagon The regular icosagon has Schläfli symbol , and can also be constructed as a truncated decagon, , or a ...

, <20/3>: Regular star polygon 20-3.svg, Regular , p₄₀ Spirolateral_1-3_126.svg,

(1,3)_126°, p₂₀ ;108° internal angles from an equiangular 4-wound

, <20/4>: Regular polygon 5 annotated.svg, Regular degenerate, r₁₀ Spirolateral 4 108.svg,

(1…4)_108°, g₅ Equiangular quadruple-pentagon1.svg, Irregular, a₁ ;90° internal angles from an equiangular 5-wound

, <20/5>: Regular polygon 4 annotated.svg, Regular degenerate, r₈ Spirolateral 5 90.svg,

(1…5)_90°, g₄ Spirolateral 1-2-3-2-1 90.svg,

(1,2,3,2,1)_90°, i₈ ;72° internal angles from an equiangular double-wound decagram, <20/6>: Regular star polygon 10-3.svg, Regular degenerate, r₂₀ Spirolateral 2 72.svg,

(1,2)_72°, p₁₀ Spirolateral 4 72.svg, Spirolateral (1…4)_72°, g₅ ;54° internal angles from an equiangular

, <20/7>: Regular star polygon 20-7.svg, Regular , r₄₀ regular_star_truncation_10-3_2.svg, Isogonal, p₂₀ regular_star_truncation_10-3_3.svg, Isogonal, p₂₀ regular_star_truncation_10-3_5.svg, Isogonal, p₂₀ ;36° internal angles from an equiangular quadruple-wound

, <20/8>: Regular star polygon 5-2.svg, Regular degenerate, r₁₀ Spirolateral_4_36.svg,

(1…4)_36°, g₅ Equiangular quadruple-pentagram1.svg, irregular, a₁ ;18° internal angles from an equiangular

, <20/9>: Regular star polygon 20-9.svg, Regular , r₄₀ regular_polygon_truncation_10_5.svg, Isogonal, p₂₀ regular_polygon_truncation_10_4.svg, Isogonal, p₂₀ regular_polygon_truncation_10_3.svg, Isogonal, p₂₀ regular_polygon_truncation_10_2.svg, Isogonal, p₂₀

References

*Williams, R. ''The Geometrical Foundation of Natural Structure: A Source Book of Design''. New York:

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...

, 1979. p. 32

External links

A Property of Equiangular Polygons: What Is It About?
a discussion of Viviani's theorem at

Cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

. * {{polygons Types of polygons