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These properties apply to all regular polygons, whether convex or star: A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon.
This image is a derivative work of the following images: File:Polygon_area_formula.jpg licensed with PD-self 2010-02-08T17:06:40Z Isalar 518x506 (20204 Bytes) ...
In either case, the area formula is correct in absolute value. This is commonly called the shoelace formula or surveyor's formula. [6] The area A of a simple polygon can also be computed if the lengths of the sides, a 1, a 2, ..., a n and the exterior angles, θ 1, θ 2, ..., θ n are known, from:
There is one regular star polygon: {12/5}, using the same vertices, but connecting every fifth point. There are also three compounds: {12/2} is reduced to 2{6} as two hexagons , and {12/3} is reduced to 3{4} as three squares , {12/4} is reduced to 4{3} as four triangles, and {12/6} is reduced to 6{2} as six degenerate digons .
Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
A regular hendecagon is represented by Schläfli symbol {11}.. A regular hendecagon has internal angles of 147. 27 degrees (=147 degrees). [5] The area of a regular hendecagon with side length a is given by [2]
A regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of an octagonal antiprism with the same D 8d , [2 + ,16] symmetry, order 32.
The area of any regular polygon is: = where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula