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The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Can tile the plane. pentagon: 5 [21] The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle. hexagon: 6
[5] The subdivision of the polygon into triangles forms a planar graph, and Euler's formula + = gives an equation that applies to the number of vertices, edges, and faces of any planar graph. The vertices are just the grid points of the polygon; there are = + of them. The faces are the triangles of the subdivision, and the single region of the ...
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, [1]: p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.
Convex hull ( in blue and yellow) of a simple polygon (in blue) The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets.
Then ,, and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle is the sum of its three components, and the angle is the sum of its two components.
A sequence of regular polygons with numbers of sides equal to powers of two, inscribed in a circle. The ratios between areas or perimeters of consecutive polygons in the sequence give the terms of Viète's formula. Viète obtained his formula by comparing the areas of regular polygons with 2 n and 2 n + 1 sides inscribed in a circle.
It states that, for given numbers of triangles, quadrilaterals, pentagons, heptagons, and other polygons other than hexagons, there exists a convex polyhedron with those given numbers of faces of each type (and an unspecified number of hexagonal faces) if and only if those numbers of polygons obey a linear equation derived from Euler's ...