Equiangular polygon
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
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 .
Examples
- An equilateral triangle is just an equilateral triangle with interior angles to and exterior angles to .
- An equiangular quadrilateral is a rectangle with inside and outside angles each .
- 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
- .
- and all exterior angles
- .
- 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)