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In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least four, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.

Examples

All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square).

Convex equilateral pentagon

Concave equilateral pentagon

A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.

A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.^[1]

Measurement

Viviani's theorem generalizes to equilateral polygons:^[2] The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.

The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there exists a principal diagonal d₁ such that^[3]

{\frac {d_{1}}{a}}\leq 2

and a principal diagonal d₂ such that

{\frac {d_{2}}{a}}>{\sqrt {3}}

.

Optimality

When an equilateral polygon is inscribed in a Reuleaux polygon, it forms a Reinhardt polygon. Among all convex polygons with the same number of sides, these polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.^[4]

References

^ De Villiers, Michael (March 2011), "Equi-angled cyclic and equilateral circumscribed polygons" (PDF), Mathematical Gazette, 95: 102–107, doi:10.1017/S0025557200002461, archived from the original (PDF) on 2016-03-03, retrieved 2015-04-29.
^ De Villiers, Michael (2012), "An illustration of the explanatory and discovery functions of proof", Leonardo, 33 (3): 1–8, doi:10.4102/pythagoras.v33i3.193, explaining (proving) Viviani's theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the 'common factor' of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon.
^ Inequalities proposed in “Crux Mathematicorum”, [1], p.184,#286.3.
^ Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata, 198: 1–18, arXiv:1405.5233, doi:10.1007/s10711-018-0326-5, MR 3933447, S2CID 119629098

External links

Media related to Equilateral polygons at Wikimedia Commons
Equilateral triangle With interactive animation
A Property of Equiangular Polygons: What Is It About? a discussion of Viviani's theorem at Cut-the-knot.

[1] De Villiers, Michael (March 2011), "Equi-angled cyclic and equilateral circumscribed polygons" (PDF), Mathematical Gazette, 95: 102–107, doi:10.1017/S0025557200002461, archived from the original (PDF) on 2016-03-03, retrieved 2015-04-29.

[2] De Villiers, Michael (2012), "An illustration of the explanatory and discovery functions of proof", Leonardo, 33 (3): 1–8, doi:10.4102/pythagoras.v33i3.193, explaining (proving) Viviani's theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the 'common factor' of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon.

[Crux-3] Inequalities proposed in “Crux Mathematicorum”, [1], p.184,#286.3.

[4] Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata, 198: 1–18, arXiv:1405.5233, doi:10.1007/s10711-018-0326-5, MR 3933447, S2CID 119629098

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[2]

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@@ Line 1: / Line 1: @@
 {{short description|Polygon in which all sides have equal length}}
 {{refimprove|date=August 2012}}
-In [[geometry]], three or more than three straight lines (or segment of a line) make a polygon and   an '''equilateral polygon''' is a [[polygon]] which has all sides of the same length. Except in the [[triangle]] case, it doesn’t need to be [[equiangular polygon|equiangular]] (doesn’t need to have all angles equal), but if it does then it is a [[regular polygon]]. If the number of sides is at least five, an equilateral polygon doesn’t need to be a [[convex polygon]]: it could be [[concave polygon|concave]] or even [[list of self-intersecting polygons|self-intersecting]].
+In [[geometry]], an '''equilateral polygon''' is a [[polygon]] which has all sides of the same length. Except in the [[triangle]] case, an equilateral polygon does not need to also be [[equiangular polygon|equiangular]] (have all angles equal), but if it does then it is a [[regular polygon]]. If the number of sides is at least four, an equilateral polygon does not need to be a [[convex polygon]]: it could be [[concave polygon|concave]] or even [[list of self-intersecting polygons|self-intersecting]].
 ==Examples==
-All [[regular polygon]]s and [[isotoxal polygon|edge-transitive polygon]]s are equilateral. When an equilateral polygon is non-crossing and [[Cyclic polygon|cyclic]] (its vertices are on a circle) it must be regular. An equilateral [[quadrilateral]] must be convex; this polygon is either a [[rhombus]] or a [[square]].
+All [[regular polygon]]s and [[isotoxal polygon|edge-transitive polygon]]s are equilateral. When an equilateral polygon is non-crossing and [[Cyclic polygon|cyclic]] (its vertices are on a circle) it must be regular. An equilateral [[quadrilateral]] must be convex; this polygon is a [[rhombus]] (possibly a [[square]]).
 {{multiple image|total_width=360|image1=5-gon equilateral 01.svg|caption1=Convex equilateral pentagon|image2=5-gon equilateral 03.svg|caption2=Concave equilateral pentagon}}
-A convex [[equilateral pentagon]] can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral pentagons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.
+A convex [[equilateral pentagon]] can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.
-A [[tangential polygon]] (one that has an [[incircle]] tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides ''n'' is odd, a tangential polygon is equilateral if and only if it is regular.<ref>{{citation|last=De Villiers|first=Michael|title=Equi-angled cyclic and equilateral circumscribed polygons|journal=[[Mathematical Gazette]]|volume=95|date=March 2011|pages=102–107|url=http://frink.machighway.com/~dynamicm/equi-anglecyclicpoly.pdf}}.</ref>
+A [[tangential polygon]] (one that has an [[incircle]] tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides ''n'' is odd, a tangential polygon is equilateral if and only if it is regular.<ref>{{citation|last=De Villiers|first=Michael|title=Equi-angled cyclic and equilateral circumscribed polygons|journal=[[Mathematical Gazette]]|volume=95|date=March 2011|pages=102–107|doi=10.1017/S0025557200002461|url=http://frink.machighway.com/~dynamicm/equi-anglecyclicpoly.pdf|access-date=2015-04-29|archive-date=2016-03-03|archive-url=https://web.archive.org/web/20160303172114/http://frink.machighway.com/~dynamicm/equi-anglecyclicpoly.pdf|url-status=dead}}.</ref>
 ==Measurement==
-[[Viviani's theorem]] generalizes to equilateral polygons:<ref>{{citation|last=De Villiers|first=Michael|title=An illustration of the explanatory and discovery functions of proof|journal=[[Leonardo (journal)|Leonardo]]|volume=33|issue=3|pages=1–8|url=http://www.pythagoras.org.za/index.php/pythagoras/article/view/193/228|quote=explaining (proving) Viviani’s theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the ‘common factor’ of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon}}.</ref> The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.
+[[Viviani's theorem]] generalizes to equilateral polygons:<ref>{{citation|last=De Villiers|first=Michael|title=An illustration of the explanatory and discovery functions of proof|journal=[[Leonardo (journal)|Leonardo]]|year=2012|volume=33|issue=3|pages=1–8|doi=10.4102/pythagoras.v33i3.193|url=http://www.pythagoras.org.za/index.php/pythagoras/article/view/193/228|quote=explaining (proving) Viviani’s theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the ‘common factor’ of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon|doi-access=free}}.</ref> The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.
 The ''principal diagonals'' of a [[hexagon]] each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side ''a'', there exists a principal diagonal ''d''<sub>1</sub> such that<ref name=Crux>''Inequalities proposed in “[[Crux Mathematicorum]]”'', [http://www.imomath.com/othercomp/Journ/ineq.pdf], p.184,#286.3.</ref>
@@ Line 21: / Line 21: @@
 and a principal diagonal ''d''<sub>2</sub> such that
-:<math>\frac{d_2}{a} > \sqrt{3}.</math>
+:<math>\frac{d_2}{a} > \sqrt{3}</math>.
 ==Optimality==
 {{main|Reinhardt polygon}}
-[[File:Reinhardt 15-gons.svg|thumb|Four Reinhardt 15-gons]]
+[[File:Reinhardt 15-gons.svg|thumb|Four Reinhardt pentadecagons]]
 When an equilateral polygon is inscribed in a [[Reuleaux polygon]], it forms a [[Reinhardt polygon]]. Among all convex polygons with the same number of sides, these polygons have the largest possible [[perimeter]] for their [[diameter]], the largest possible [[Curve of constant width|width]] for their diameter, and the largest possible width for their perimeter.<ref>{{citation
  | last1 = Hare | first1 = Kevin G.
@@ Line 36: / Line 36: @@
  | volume = 198
  | year = 2019| arxiv = 1405.5233
+ | s2cid = 119629098
  }}</ref>