A regular star polygon joins every -th vertex of a regular -gon. When , this traces one closed polygonal circuit.
For a simple polygon with cyclically ordered vertices of a polygon , its signed area is , with indices read cyclically. Reversing the traversal changes the sign; taking the absolute value gives the area. Each summand is a two-dimensional determinant.
Positive real number lengths , with , are the sides of a nondegenerate polygon exactly when . Necessity follows from the triangle inequality: the longest side must be shorter than the path formed by the remaining sides.
The sum of the lengths of its sides of a polygon. For a triangle with side lengths , the perimeter of a polygon is .
A polygon with five sides and five vertices. A regular pentagon has equal sides and angles; its symmetry group is the order-ten dihedral group, consisting of five rotations and five reflections. Each reflection axis passes through a vertex and the midpoint of the opposite edge.
A polygon with six sides and six vertices. A regular hexagon has equal sides and angles, with each interior angle equal to . Its symmetry group consists of six rotations and six reflections, forming a dihedral group of order twelve.
A side of a polygon is a line segment joining two consecutive vertices of a polygon in its boundary.
A diagonal of a polygon joins two vertices of a polygon that are not endpoints of the same side of a polygon. A square has two diagonals of a polygon, forming a two-element group orbit under its dihedral group of symmetries.
A Euclidean polygon with four sides and four right angles. Opposite sides are parallel and have equal lengths. A rectangle is a square exactly when its adjacent side lengths agree.
A rectangle with all four side lengths equal. It is a regular four-sided polygon. Rotations and translations preserve this definition; its sides need not be parallel to the coordinate axes.
A square frame formed by removing a concentric square hole from a larger square. If outer and inner sides are positive integers with the same integer parity, the border has integer thickness and contains unit tiles.
A parallelogram is the convex set with vertices for two linearly independent planar vectors . Its opposite edges are parallel and its centre gives central symmetry. Its area is , so an invertible linear map changes its area by the absolute value of that map's determinant. These facts make parallelograms convenient convex bodies for two-dimensional applications of the Minkowski convex body theorem.
For a quadrilateral with side lengths , semiperimeter , area , and opposite angles ,The second term vanishes exactly when , which is the cyclicity condition for a nondegenerate convex quadrilateral.
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Polygon is a protocol and framework for building and connecting Ethereum-compatible blockchain networks. It seeks to address some of the scalability issues faced by the Ethereum network by enabling the creation of Layer 2 scaling solutions. Originally known as Matic Network, it rebranded to Polygon in early 2021.
