geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an equilateral 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 ...

which has all sides of the same length. Except in the

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 ...

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 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 ...

. If the number of sides is at least four, an equilateral polygon does not need to be a

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

: it could be concave or even self-intersecting.

Examples

All

s 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 In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

must be convex; this polygon is a

rhombus 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 ...

(possibly a

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 ...

). 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.

Measurement

Viviani's theorem generalizes to equilateral polygons: 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''₁ such that''Inequalities proposed in “

Crux Mathematicorum ''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. Its editor-in-chief is Kseniya Garaschuk. The journ ...

”''

p.184,#286.3. :

\frac \leq 2

and a principal diagonal ''d''₂ such that :

\frac > \sqrt

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 A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

for their

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

, the largest possible

width Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Intern ...

for their diameter, and the largest possible width for their perimeter.

References

External links

*
Equilateral triangle
With interactive animation
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 ...

. Types of polygons {{polygons