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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 .
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For clarity, a planar equiangular polygon can be called direct or ...
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex , star or skew .
The polygon is also cyclic and equiangular. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential. The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral.
If a polygon is regular (both equiangular and equilateral), the sum of the distances to the sides from an interior point is independent of the location of the point. Specifically, it equals n times the apothem , where n is the number of sides and the apothem is the distance from the center to a side.
A regular star polygon is a self-intersecting, equilateral, and equiangular polygon. A regular star polygon is denoted by its Schläfli symbol { p / q }, where p (the number of vertices) and q (the density ) are relatively prime (they share no factors) and where q ≥ 2.
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Some Reuleaux polygons have side lengths that are irrational multiples of each other, but if a Reuleaux polygon has sides that can be partitioned into a system of arcs of equal length, then the polygon formed as the convex hull of the endpoints of these arcs is defined as a Reinhardt polygon. Necessarily, the vertices of the underlying Reuleaux ...
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