{"history":[],"article":{"id":"e299436c-b4b0-4b52-b0ec-18d365ba5023","title":"Equiangular polygon","slug":"equiangular-polygon","content":":::summary\n:::\n\n# \nIn [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry), an **equiangular [polygon](https://bliptext.com/articles/polygon)** is a [polygon](https://bliptext.com/articles/polygon) whose vertex angles are equal.\nIf the lengths of the sides are also equal (that is, if it is also [equilateral](https://bliptext.com/articles/equilateral-polygon)) then it is a [regular polygon](https://bliptext.com/articles/regular-polygon).\n[Isogonal polygons](https://bliptext.com/articles/isogonal-polygon) are equiangular polygons which alternate two edge lengths.\nFor clarity, a **planar equiangular polygon** can be called direct or indirect.\nA **direct equiangular polygon** has all angles turning in the same direction in a plane and can include multiple [turns](https://bliptext.com/articles/turn-angle).\n**Convex equiangular polygons** are always direct.\nAn **indirect equiangular polygon** can include angles turning right or left in any combination.\nA **[skew](https://bliptext.com/articles/skew-polygon) equiangular polygon** may be [isogonal](https://bliptext.com/articles/isogonal-figure), but can't be considered direct since it is nonplanar.\nA [spirolateral](https://bliptext.com/articles/spirolateral) nθ 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.\n\n# Construction\nAn equiangular polygon can be constructed from a [regular polygon](https://bliptext.com/articles/regular-polygon) or [regular star polygon](https://bliptext.com/articles/regular-star-polygon) where edges are extended as infinite [lines](https://bliptext.com/articles/line-geometry).\nEach edges can be independently moved perpendicular to the line's direction.\nVertices represent the intersection point between pairs of neighboring line.\nEach moved line adjusts its edge-length and the lengths of its two neighboring edges.\nIf edges are reduced to zero length, the polygon becomes degenerate, or if reduced to negative lengths, this will reverse the internal and external angles.\nFor an even-sided direct equiangular polygon, with internal angles θ°, moving alternate edges can invert all vertices into [supplementary angles](https://bliptext.com/articles/supplementary-angle), 180-θ°.\nOdd-sided direct equiangular polygons can only be partially inverted, leaving a mixture of supplementary angles.\nEvery equiangular polygon can be adjusted in proportions by this construction and still preserve equiangular status.\n\n# Equiangular polygon theorem\nFor a convex equiangular p-gon, each [internal angle](https://bliptext.com/articles/internal-angle) is 180(1-2/p)°; this is the equiangular polygon theorem.\nFor a direct equiangular p/q [star polygon](https://bliptext.com/articles/star-polygon), [density](https://bliptext.com/articles/density-geometry) q, each internal angle is 180(1-2q/p)°, with 1<2q1, this represents a w-wound (p/w)/(q/w) star polygon, which is degenerate for the regular case.\nA concave indirect equiangular (pr+pl)-gon, with pr right turn vertices and pl left turn vertices, will have internal angles of 180(1-2/|pr-pl|))°, regardless of their sequence. An indirect star equiangular (pr+pl)-gon, with pr right turn vertices and pl left turn vertices and q total [turns](https://bliptext.com/articles/turn-angle), will have internal angles of 180(1-2q/|pr-pl|))°, regardless of their sequence.\nAn equiangular polygon with the same number of right and left turns has zero total turns, and has no constraints on its angles.\n\n# Notation\nEvery direct equiangular p-gon can be given a notation <p> or <p/q>, like [regular polygons](https://bliptext.com/articles/regular-polygon) {p} and [regular star polygons](https://bliptext.com/articles/regular-star-polygon) {p/q}, containing p vertices, and stars having [density](https://bliptext.com/articles/density-polytope) q.\nConvex equiangular p-gons <p> have internal angles 180(1-2/p)°, while direct star equiangular polygons, <p/q>, have internal angles 180(1-2q/p)°.\nA concave indirect equiangular p-gon can be given the notation , with c counter-turn vertices.\nFor example, <6-2> is a hexagon with 90° internal angles of the difference, <4>, 1 counter-turned vertex.\nA multiturn indirect equilateral p-gon can be given the notation with c counter turn vertices, and q total [turns](https://bliptext.com/articles/turn-angle).\nAn equiangular polygon is a p-gon with undefined internal angles θ, but can be expressed explicitly as θ.\n\n# Other properties\n[Viviani's theorem](https://bliptext.com/articles/viviani-s-theorem) holds for equiangular polygons:\nA [cyclic polygon](https://bliptext.com/articles/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).\nThus if n is odd, a cyclic polygon is equiangular if and only if it is regular.\nFor prime p, every integer-sided equiangular p-gon is regular.\nMoreover, every integer-sided equiangular pk-gon has p-fold [rotational symmetry](https://bliptext.com/articles/rotational-symmetry).\nAn 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\n\n# Direct equiangular polygons by sides\nDirect equiangular polygons can be regular, isogonal, or lower symmetries.\nExamples for <p/q> are grouped into sections by p and subgrouped by density q.\n\n## Equiangular triangles\n**Equiangular triangles** must be convex and have 60° internal angles.\nIt is an [equilateral triangle](https://bliptext.com/articles/equilateral-triangle) and a [regular triangle](https://bliptext.com/articles/regular-triangle), <3>={3}.\nThe only degree of freedom is edge-length.\n\n## Equiangular quadrilaterals\nDirect **equiangular quadrilaterals** have 90° internal angles.\nThe only equiangular quadrilaterals are [rectangles](https://bliptext.com/articles/rectangle), <4>, and [squares](https://bliptext.com/articles/square), {4}.\nAn equiangular quadrilateral with integer side lengths may be tiled by [unit squares](https://bliptext.com/articles/unit-square).\n# [A rectangle dissected into a 2×3 array of squares | upload.wikimedia.org/wikipedia/commons/a/a3/Rectangle-2x3.svg]\n\n## Equiangular pentagons\nDirect **equiangular pentagons**, <5> and <5/2>, have 108° and 36° internal angles respectively.\nEquiangular pentagons can be [regular](https://bliptext.com/articles/regular-pentagon), have bilateral symmetry, or no symmetry.\n\n## Equiangular hexagons\nDirect **equiangular hexagons**, <6> and <6/2>, have 120° and 60° internal angles respectively.\nAn equiangular [hexagon](https://bliptext.com/articles/hexagon) with integer side lengths may be tiled by unit [equilateral triangles](https://bliptext.com/articles/equilateral-triangle).\n# [An equiangular hexagon with 1:2 edge length ratios, with equilateral triangles. This is spirolateral 2120°. | upload.wikimedia.org/wikipedia/commons/0/05/Equiangular///hexagon-1-2.svg]\n\n## Equiangular heptagons\nDirect **equiangular heptagons**, <7>, <7/2>, and <7/3> have 128 4/7°, 77 1/7° and 25 5/7° internal angles respectively.\n\n## Equiangular octagons\nDirect **equiangular octagons**, <8>, <8/2> and <8/3>, have 135°, 90° and 45° internal angles respectively.\n\n## Equiangular enneagons\nDirect **equiangular enneagons**, <9>, <9/2>, <9/3>, and <9/4> have 140°, 100°, 60° and 20° internal angles respectively.\n\n## Equiangular decagons\nDirect **equiangular decagons**, <10>, <10/2>, <10/3>, <10/4>, have 144°, 108°, 72° and 36° internal angles respectively.\n\n## Equiangular hendecagons\nDirect **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.\n\n## Equiangular dodecagons\nDirect **equiangular dodecagons**, <12>, <12/2>, <12/3>, <12/4>, and <12/5> have 150°, 120°, 90°, 60°, and 30° internal angles respectively.\nConvex solutions with integer edge lengths may be tiled by [pattern blocks](https://bliptext.com/articles/pattern-blocks), squares, equilateral triangles, and 30° [rhombi](https://bliptext.com/articles/rhombus).\n\n## Equiangular tetradecagons\nDirect **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.\n\n## Equiangular pentadecagons\nDirect **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.\n\n## Equiangular hexadecagons\nDirect **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.\n\n## Equiangular octadecagons\nDirect **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.\n\n## Equiangular icosagons\nDirect **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.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:06:15.252Z","updated_at":"2025-01-25T22:06:15.252Z","search_vector":"'equiangular':1 'polygon':2","search_content":" \n \n\n# \nIn [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry), an **equiangular [polygon](https://bliptext.com/articles/polygon)** is a [polygon](https://bliptext.com/articles/polygon) whose vertex angles are equal.\nIf the lengths of the sides are also equal (that is, if it is also [equilateral](https://bliptext.com/articles/equilateral-polygon)) then it is a [regular polygon](https://bliptext.com/articles/regular-polygon).\n[Isogonal polygons](https://bliptext.com/articles/isogonal-polygon) are equiangular polygons which alternate two edge lengths.\nFor clarity, a **planar equiangular polygon** can be called direct or indirect.\nA **direct equiangular polygon** has all angles turning in the same direction in a plane and can include multiple [turns](https://bliptext.com/articles/turn-angle).\n**Convex equiangular polygons** are always direct.\nAn **indirect equiangular polygon** can include angles turning right or left in any combination.\nA **[skew](https://bliptext.com/articles/skew-polygon) equiangular polygon** may be [isogonal](https://bliptext.com/articles/isogonal-figure), but can't be considered direct since it is nonplanar.\nA [spirolateral](https://bliptext.com/articles/spirolateral) nθ 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.\n\n# Construction\nAn equiangular polygon can be constructed from a [regular polygon](https://bliptext.com/articles/regular-polygon) or [regular star polygon](https://bliptext.com/articles/regular-star-polygon) where edges are extended as infinite [lines](https://bliptext.com/articles/line-geometry).\nEach edges can be independently moved perpendicular to the line's direction.\nVertices represent the intersection point between pairs of neighboring line.\nEach moved line adjusts its edge-length and the lengths of its two neighboring edges.\nIf edges are reduced to zero length, the polygon becomes degenerate, or if reduced to negative lengths, this will reverse the internal and external angles.\nFor an even-sided direct equiangular polygon, with internal angles θ°, moving alternate edges can invert all vertices into [supplementary angles](https://bliptext.com/articles/supplementary-angle), 180-θ°.\nOdd-sided direct equiangular polygons can only be partially inverted, leaving a mixture of supplementary angles.\nEvery equiangular polygon can be adjusted in proportions by this construction and still preserve equiangular status.\n\n# Equiangular polygon theorem\nFor a convex equiangular p-gon, each [internal angle](https://bliptext.com/articles/internal-angle) is 180(1-2/p)°; this is the equiangular polygon theorem.\nFor a direct equiangular p/q [star polygon](https://bliptext.com/articles/star-polygon), [density](https://bliptext.com/articles/density-geometry) q, each internal angle is 180(1-2q/p)°, with 1 1, this represents a w-wound (p/w)/(q/w) star polygon, which is degenerate for the regular case.\nA concave indirect equiangular (pr+pl)-gon, with pr right turn vertices and pl left turn vertices, will have internal angles of 180(1-2/|pr-pl|))°, regardless of their sequence. An indirect star equiangular (pr+pl)-gon, with pr right turn vertices and pl left turn vertices and q total [turns](https://bliptext.com/articles/turn-angle), will have internal angles of 180(1-2q/|pr-pl|))°, regardless of their sequence.\nAn equiangular polygon with the same number of right and left turns has zero total turns, and has no constraints on its angles.\n\n# Notation\nEvery direct equiangular p-gon can be given a notation <p> or <p/q>, like [regular polygons](https://bliptext.com/articles/regular-polygon) {p} and [regular star polygons](https://bliptext.com/articles/regular-star-polygon) {p/q}, containing p vertices, and stars having [density](https://bliptext.com/articles/density-polytope) q.\nConvex equiangular p-gons <p> have internal angles 180(1-2/p)°, while direct star equiangular polygons, <p/q>, have internal angles 180(1-2q/p)°.\nA concave indirect equiangular p-gon can be given the notation , with c counter-turn vertices.\nFor example, is a hexagon with 90° internal angles of the difference, , 1 counter-turned vertex.\nA multiturn indirect equilateral p-gon can be given the notation with c counter turn vertices, and q total [turns](https://bliptext.com/articles/turn-angle).\nAn equiangular polygon is a p-gon with undefined internal angles θ, but can be expressed explicitly as θ.\n\n# Other properties\n[Viviani's theorem](https://bliptext.com/articles/viviani-s-theorem) holds for equiangular polygons:\nA [cyclic polygon](https://bliptext.com/articles/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).\nThus if n is odd, a cyclic polygon is equiangular if and only if it is regular.\nFor prime p, every integer-sided equiangular p-gon is regular.\nMoreover, every integer-sided equiangular pk-gon has p-fold [rotational symmetry](https://bliptext.com/articles/rotational-symmetry).\nAn 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\n\n# Direct equiangular polygons by sides\nDirect equiangular polygons can be regular, isogonal, or lower symmetries.\nExamples for <p/q> are grouped into sections by p and subgrouped by density q.\n\n## Equiangular triangles\n**Equiangular triangles** must be convex and have 60° internal angles.\nIt is an [equilateral triangle](https://bliptext.com/articles/equilateral-triangle) and a [regular triangle](https://bliptext.com/articles/regular-triangle), ={3}.\nThe only degree of freedom is edge-length.\n\n## Equiangular quadrilaterals\nDirect **equiangular quadrilaterals** have 90° internal angles.\nThe only equiangular quadrilaterals are [rectangles](https://bliptext.com/articles/rectangle), , and [squares](https://bliptext.com/articles/square), {4}.\nAn equiangular quadrilateral with integer side lengths may be tiled by [unit squares](https://bliptext.com/articles/unit-square).\n# [A rectangle dissected into a 2×3 array of squares | upload.wikimedia.org/wikipedia/commons/a/a3/Rectangle-2x3.svg]\n\n## Equiangular pentagons\nDirect **equiangular pentagons**, and , have 108° and 36° internal angles respectively.\nEquiangular pentagons can be [regular](https://bliptext.com/articles/regular-pentagon), have bilateral symmetry, or no symmetry.\n\n## Equiangular hexagons\nDirect **equiangular hexagons**, and , have 120° and 60° internal angles respectively.\nAn equiangular [hexagon](https://bliptext.com/articles/hexagon) with integer side lengths may be tiled by unit [equilateral triangles](https://bliptext.com/articles/equilateral-triangle).\n# [An equiangular hexagon with 1:2 edge length ratios, with equilateral triangles. This is spirolateral 2120°. | upload.wikimedia.org/wikipedia/commons/0/05/Equiangular///hexagon-1-2.svg]\n\n## Equiangular heptagons\nDirect **equiangular heptagons**, , , and have 128 4/7°, 77 1/7° and 25 5/7° internal angles respectively.\n\n## Equiangular octagons\nDirect **equiangular octagons**, , and , have 135°, 90° and 45° internal angles respectively.\n\n## Equiangular enneagons\nDirect **equiangular enneagons**, , , , and have 140°, 100°, 60° and 20° internal angles respectively.\n\n## Equiangular decagons\nDirect **equiangular decagons**, , , , , have 144°, 108°, 72° and 36° internal angles respectively.\n\n## Equiangular hendecagons\nDirect **equiangular hendecagons**, , , , , and have 147 3/11°, 114 6/11°, 81 9/11°, 49 1/11°, and 16 4/11° internal angles respectively.\n\n## Equiangular dodecagons\nDirect **equiangular dodecagons**, , , , , and have 150°, 120°, 90°, 60°, and 30° internal angles respectively.\nConvex solutions with integer edge lengths may be tiled by [pattern blocks](https://bliptext.com/articles/pattern-blocks), squares, equilateral triangles, and 30° [rhombi](https://bliptext.com/articles/rhombus).\n\n## Equiangular tetradecagons\nDirect **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.\n\n## Equiangular pentadecagons\nDirect **equiangular pentadecagons**, , , , , , , and , have 156°, 132°, 108°, 84°, 60° and 12° internal angles respectively.\n\n## Equiangular hexadecagons\nDirect **equiangular hexadecagons**, , , , , , , and , have 157.5°, 135°, 112.5°, 90°, 67.5° 45° and 22.5° internal angles respectively.\n\n## Equiangular octadecagons\nDirect **equiangular octadecagons**, , , , , , , and , have 160°, 140°, 120°, 100°, 80°, 60°, 40° and 20° internal angles respectively.\n\n## Equiangular icosagons\nDirect **equiangular icosagon**, , , , , , , and , have 162°, 126°, 108°, 90°, 72°, 54° and 18° internal angles respectively.","revisionCount":0},"hasMore":false,"nextCursor":null}

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