Equiangular polygon

Equiangular hexagons

An equiangular polygon in the geometry of a polygon of the Euclidean plane in which all interior angles are equal. Equiangular polygons are to be distinguished from equilateral polygons , in which the polygon sides are all of the same length. A polygon that is both equiangular and equilateral is called a regular polygon .

definition

A polygon is called equiangular if the interior angles of the polygon are all the same size, that is, if ${\ displaystyle \ alpha, \ beta, \ gamma, \ ldots}$

{\ displaystyle \ alpha = \ beta = \ gamma = \ ldots}

applies. Since the inside and outside angles at the corners of a polygon add up to 180 °, all outside angles are equivalent to this in an equiangular polygon . ${\ displaystyle \ alpha ', \ beta', \ gamma ', \ ldots}$

Examples

An equilateral triangle is just an equilateral triangle with interior angles to and exterior angles to . ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle 120 ^ {\ circ}}$
An equiangular quadrilateral is a rectangle with inside and outside angles each . ${\ displaystyle 90 ^ {\ circ}}$
A regular polygon is an equiangular polygon that is also equilateral.

properties

A tangent polygon that is equiangular is always equilateral and therefore regular.
A chord polygon is equiangular if and only if the side lengths alternate between two values.
A simple, that is, a non- overturned , equiangular polygon is always convex . Since the sum of the angles in a simple corner always results, measure all interior angles in a simple equiangular polygon ${\ displaystyle n}$ ${\ displaystyle (n-2) \ cdot 180 ^ {\ circ}}$

{\ displaystyle \ alpha = \ beta = \ gamma = \ ldots = {\ frac {n-2} {n}} \ cdot 180 ^ {\ circ}}

.

and all exterior angles

{\ displaystyle \ alpha '= \ beta' = \ gamma '= \ ldots = {\ frac {1} {n}} \ cdot 360 ^ {\ circ}}

.

In simple equiangular polygons, Viviani's theorem also applies , according to which the sum of the distances from any point inside the polygon to the sides of the polygon is independent of the position of the point.

Individual evidence

↑ Michael De Villiers: Equiangular cyclic and equilateral polygons circumscribed . In: Mathematical Gazette . No. 95 , 2011, pp. 102-107 .

Web links

Eric W. Weisstein : Equiangular Polygon . In: MathWorld (English).
Wkbj79: Equiangular Polygon . In: PlanetMath . (English)

[1] Michael De Villiers: Equiangular cyclic and equilateral polygons circumscribed . In: Mathematical Gazette . No. 95 , 2011, pp. 102-107 .