In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, an equiangular polygon is a
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also
equilateral
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
) 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 ...
.
Isogonal polygons 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 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, ...
''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 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 ...
or
regular star 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 ...
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 angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
s, 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 adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
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, Decagram (geometry)#Related figures, certain notable ones can ...
,
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
''q'', each internal angle is 180(1−2''q''/''p'')°, with . For , this represents a ''w''-wound star polygon, which is degenerate for the regular case.
A concave ''indirect equiangular'' -gon, with right turn vertices and left turn vertices, will have internal angles of , regardless of their sequence. An ''indirect star equiangular'' -gon, with right turn vertices and left turn vertices and ''q'' total
turns, will have internal angles of , 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 or , like
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 ...
s and
regular star 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 ...
s , containing ''p'' vertices, and stars having
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
''q''.
Convex equiangular ''p''-gons have internal angles 180(1−2/''p'')°, while direct star equiangular polygons, , have internal angles 180(1−2''q''/''p'')°.
A concave indirect equiangular ''p''-gon can be given the notation , with ''c'' counter-turn vertices. For example, is a hexagon with 90° internal angles of the difference, , 1 counter-turned vertex. A multiturn indirect equilateral ''p''-gon can be given the notation with ''c'' counter turn vertices, and ''q'' total
turns. An equiangular polygon <''p''−''p''> is a ''p''-gon with undefined internal angles , but can be expressed explicitly as
''θ''.
Other properties
Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest 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 ...
holds for equiangular polygons:
[Elias Abboud "On Viviani's Theorem and its Extensions"](_blank)
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, a set (mathematics), set of point (geometry), points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertex (geometry), vertices are concyclic is called a cyclic polygon, and the circle is cal ...
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 (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
.
An ordered set of side lengths
gives rise to an equiangular ''n''-gon if and only if either of two equivalent conditions holds for the polynomial
it equals zero at the complex value
it is divisible by
[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
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
and a
regular triangle, =. 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 geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
s, , and
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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 coordinat ...
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, ...
290°, p4
Equiangular pentagons
Direct equiangular pentagons, and , have 108° and 36° internal angles respectively.
; 108° internal angle from an equiangular
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
,
Equiangular pentagons 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 around ...
,
File:Regular star polygon 5-2.svg, Regular 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 around ...
, r10
Equiangular_pentagram1.svg, Irregular, d2
Equiangular hexagons

Direct equiangular hexagons, and , have 120° and 60° internal angles respectively.
; 120° internal angles of an equiangular
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
, :
An equiangular
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
with integer side lengths may be tiled by unit
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s.
Regular polygon 6 annotated.svg, Regular, , r12
Spirolateral_2_120.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, ...
(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, :
Regular polygon 3 annotated.svg, Regular, degenerate, r6
Spirolateral 1-3 60.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, ...
(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, , , and 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 ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...
, :
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 polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
, :
Regular star polygon 7-2.svg, Regular, r14
Equiangular heptagram1.svg, Irregular, i2
; 25.71° internal angles of an equiangular
heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
, :
Regular star polygon 7-3.svg, Regular, r14
Equiangular heptagram2.svg, Irregular, i2
Equiangular octagons
Direct equiangular octagons, , and , have 135°, 90° and 45° internal angles respectively.
; 135° internal angles from an equiangular
octagon
In geometry, an octagon () 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, which alternates two types of edges. A truncated octagon, t is a ...
, :
Regular polygon 8 annotated.svg, Regular, r16
Spirolateral_2_135.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, ...
(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
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, :
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, ''wikt:octa-, octa-'', with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''-gram'' suffix derives from γραμμή ...
, :
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, 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, ...
, (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, , , , and have 140°, 100°, 60° and 20° internal angles respectively.
;140° internal angles from an equiangular enneagon
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, :
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 3100°, g3
;60° internal angles from an equiangular ''triple-wound triangle'', :
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, :
Regular star polygon 9-4.svg, Regular , r18
Spirolateral 3 20.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, ...
320°, g3
Equiangular enneagram2.svg, Irregular, i2
Equiangular decagons
Direct equiangular decagons, , , , , have 144°, 108°, 72° and 36° internal angles respectively.
;144° internal angles from an equiangular
decagon
Regular polygon 10 annotated.svg, Regular, r20
Spirolateral 2 144.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, ...
(1,2)144°, p10
Spirolateral_5_144.svg, Spirolateral (1…5)144°, g2
;108° internal angles from an equiangular double-wound
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
Regular polygon 5 annotated.svg, Regular, degenerate
Spirolateral 2 108.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, ...
(1,2)108°, p10
Equiangular_double-pentagon1.svg, Irregular, p2
;72° internal angles from an equiangular
decagram
Regular star polygon 10-3.svg, Regular , r20
Regular star truncation 5-3 3.svg, Isogonal, p10
Spirolateral_2_72.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, ...
(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
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 around ...
Regular star polygon 5-2.svg, Regular, degenerate, r10
Spirolateral 2 36.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, ...
(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, , , , , and 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, :
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 Vertex (geometry), vertices.
The name ''hendecagram'' combines a Greek numeral prefix, ''wikt:hendeca-, hendeca-'', with the Greek language, Greek suffix ...
, :
Regular star polygon 11-2.svg, Regular , r22
;81° internal angles from an equiangular
hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven Vertex (geometry), vertices.
The name ''hendecagram'' combines a Greek numeral prefix, ''wikt:hendeca-, hendeca-'', with the Greek language, Greek suffix ...
, :
Regular star polygon 11-3.svg, Regular , r22
;49° internal angles from an equiangular
hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven Vertex (geometry), vertices.
The name ''hendecagram'' combines a Greek numeral prefix, ''wikt:hendeca-, hendeca-'', with the Greek language, Greek suffix ...
, :
Regular star polygon 11-4.svg, Regular , r22
;16° internal angles from an equiangular
hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven Vertex (geometry), vertices.
The name ''hendecagram'' combines a Greek numeral prefix, ''wikt:hendeca-, hendeca-'', with the Greek language, Greek suffix ...
, :
Regular star polygon 11-5.svg, Regular , r22
Equiangular dodecagons
Direct equiangular dodecagons, , , , , and 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 polygon, 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 ...
, :
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 ...
, squares, equilateral triangles, and 30°
rhombi
In plane Euclidean geometry, a rhombus (: 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 rhom ...
.
Regular polygon 12 annotated.svg, Regular, , r24
Regular truncation 6 0.45.svg, Isogonal, p12
Spirolateral_2_150.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, ...
(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
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
'',
Regular polygon 6 annotated.svg, Regular degenerate, r12
Spirolateral 4 120.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, ...
, (1…4)120°, g3
Equiangular double-hexagon3.svg, Irregular, d2
Equiangular double-hexagon2.svg, Irregular, d2
; 90° internal angles from an equiangular ''triple-wound
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
'',
Regular polygon 4 annotated.svg, Regular, degenerate, r8
Spirolateral_3_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, ...
(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 corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
'',
Regular polygon 3 annotated.svg, Regular, degenerate, r6
Equiangular double-hexagon4.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, ...
(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 ''
'',
Regular star polygon 12-5.svg, Regular , r24
Regular truncation 6 1.5.svg, Isogonal, p12
Spirolateral_2_30.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, ...
(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, , , , , and , , 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, :
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
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 ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...
, :
Regular polygon 7 annotated.svg, Regular degenerate, r14
regular_star_truncation_7-5_3.svg, Isogonal, t, p14
Spirolateral_2_129.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, ...
2128.57°
;102.85° internal angles from an equiangular
tetradecagram, :
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
heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
:
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, 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, ...
277.14°
;51.43° internal angles from an equiangular
tetradecagram, :
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
heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
, :
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, , , , , , , and , have 156°, 132°, 108°, 84°, 60° and 12° internal angles respectively.
;156° internal angles from an equiangular pentadecagon, :
Regular polygon 15 annotated.svg, Regular, , r30
;132° internal angles from an equiangular
pentadecagram, :
Regular star polygon 15-2.svg, Regular, , r30
;108° internal angles from an equiangular triple-wound pentagon, :
Regular polygon 5 annotated.svg, Regular, degenerate, r10
Equiangular_triple-pentagon1.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, ...
(1…3)108°, g5
;84° internal angles from an equiangular pentadecagram, :
Regular star polygon 15-4.svg, Regular, , r30
;60° internal angles from an equiangular 5-wound
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, :
Regular polygon 3 annotated.svg, Regular, degenerate, r6
Equiangular 5-wound triangle1.svg, Irregular, a1
;36° internal angles from an equiangular triple-wound
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 around ...
, :
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, :
Regular star polygon 15-7.svg, Regular, , r30
Equiangular hexadecagons
Direct equiangular hexadecagons, , , , , , , and , have 157.5°, 135°, 112.5°, 90°, 67.5° 45° and 22.5° internal angles respectively.
;157.5° internal angles from an equiangular
hexadecagon
In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon.
Regular hexadecagon
A ''regular polygon, regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. It ...
, :
Regular polygon 16 annotated.svg, Regular, , r32
File:Regular truncation 8 0.45.svg, Isogonal, t, p16
File:Spirolateral 4 1575.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, ...
(1…4)157.5°, g4
;135° internal angles from an equiangular double-wound octagon, :
Regular polygon 8 annotated.svg, Regular, degenerate, r16
Equiangular double octagon1.svg, Irregular, p16
;112.5° internal angles from an equiangular
hexadecagram, :
Regular star polygon 16-3.svg, Regular, , r32
;90° internal angles from an equiangular 4-wound square, :
Regular polygon 4 annotated.svg, Regular, degenerate, r8
Equiangular 4-wound square1.svg, Irregular, a1
;67.5° internal angles from an equiangular
hexadecagram, :
Regular star polygon 16-5.svg, Regular, , r32
;45° internal angles from an equiangular double-wound regular
octagram
In geometry, an octagram is an eight-angled star polygon.
The name ''octagram'' combine a Greek numeral prefix, ''wikt:octa-, octa-'', with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''-gram'' suffix derives from γραμμή ...
, :
Regular star polygon 8-3.svg, Regular, degenerate, r16
Spirolateral 3 45.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, ...
(1…3)45°, g8
;22.5° internal angles from an equiangular
hexadecagram, :
Regular star polygon 16-7.svg, Regular, , r32
File:Regular truncation 8 -10.svg, Isogonal, p16
Equiangular octadecagons
Direct equiangular octadecagons, <18}, , , , , , , and , 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 polygon, regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular Truncation (geometry), trunc ...
, :
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 ...
, :
Regular polygon 9.svg, Regular, degenerate
Spirolateral 2 140.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, ...
2140°, p18
; 120° internal angles of an equiangular 3-wound hexagon :
Regular polygon 6.svg, Regular, degenerate, r18
Equilateral triple-wound-hexagon1.svg, irregular, a1
; 100° internal angles of an equiangular double-wound
enneagram :
Regular star polygon 9-2.svg, Regular, degenerate, r18
File:Spirolateral_6_100.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, ...
2100°, 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 :
Regular polygon 3.svg, Regular, degenerate, r6
Equilateral 6-wound-triangle1.svg, irregular, a1
; 40° internal angles of an equiangular
octadecagram :
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 :
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, 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, ...
220°, p18
File:Spirolateral 6 20.svg, Spirolateral 620°, g3
Equiangular icosagons
Direct equiangular icosagon, , , , , , , and , have 162°, 126°, 108°, 90°, 72°, 54° and 18° internal angles respectively.
;162° internal angles from an equiangular
icosagon
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 polygon, regular icosagon has Schläfli symbol , and can also be constructed as a Truncation ( ...
, :
Regular polygon 20 annotated.svg, Regular, , r40
Spirolateral 1-3 162.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, ...
(1,3)162°, p20
;144° internal angles from an equiangular double-wound
decagon, :
Regular polygon 10 annotated.svg, Regular, degenerate, r20
Spirolateral_4_144.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, ...
(1…4)144°, g5
;126° internal angles from an equiangular
icosagram, :
Regular star polygon 20-3.svg, Regular , p40
Spirolateral_1-3_126.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, ...
(1,3)126°, p20
;108° internal angles from an equiangular 4-wound
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, :
Regular polygon 5 annotated.svg, Regular degenerate, r10
Spirolateral 4 108.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, ...
(1…4)108°, g5
Equiangular quadruple-pentagon1.svg, Irregular, a1
;90° internal angles from an equiangular 5-wound
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, :
Regular polygon 4 annotated.svg, Regular degenerate, r8
Spirolateral 5 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, ...
(1…5)90°, g4
Spirolateral 1-2-3-2-1 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, ...
(1,2,3,2,1)90°, i8
;72° internal angles from an equiangular double-wound
decagram, :
Regular star polygon 10-3.svg, Regular degenerate, r20
Spirolateral 2 72.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, ...
(1,2)72°, p10
Spirolateral 4 72.svg, Spirolateral (1…4)72°, g5
;54° internal angles from an equiangular
icosagram, :
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
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 around ...
, :
Regular star polygon 5-2.svg, Regular degenerate, r10
Spirolateral_4_36.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, ...
(1…4)36°, g5
Equiangular quadruple-pentagram1.svg, irregular, a1
;18° internal angles from an equiangular
icosagram, :
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
See also
*
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, ...
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, book ...
, 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 Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
.
*
{{polygons
Types of polygons