Regular polygon

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
Regular pentagon (n = 5) with side s, circumradius R and apothem a
Comparison of sizes of regular polygons with the same edge length, from three to sixty sides. The size increases without bound as the number of sides approaches infinity.

Polygon that is direct equiangular and equilateral (all sides have the same length).

- Regular polygon

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Circumscribed circle

Circle that passes through all the vertices of the polygon.

Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
Construction of the circumcircle of triangle ABC (\Delta ABC) and the circumcenter Q
Alternative construction of the circumcenter (intersection of broken lines)
Cyclic quadrilaterals
A sequence of circumscribed polygons and circles.

All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic.

Polygon

Plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain .

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.
Some different types of polygon
Partitioning an n-gon into n − 2 triangles
Coordinates of a non-convex pentagon.
Historical image of polygons (1699)
The Giant's Causeway, in Northern Ireland

Regular: both equilateral and equiangular.

Dihedral group

The symmetry group of a snowflake is D6, a dihedral symmetry, the same as for a regular hexagon.
The six axes of reflection of a regular hexagon
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The lines of reflection labelled S0, S1, and S2 remain fixed in space (on the page) and do not themselves move when a symmetry operation (rotation or reflection) is done on the triangle (this matters when doing compositions of symmetries).
The composition of these two reflections is a rotation.
The symmetries of this pentagon are linear transformations of the plane as a vector space.
Example subgroups from a hexagonal dihedral symmetry
The four elements of D2 (x-axis is vertical here)
D4 is nonabelian (x-axis is vertical here).
2D D{{sub|16}} symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals.
2D D{{sub|6}} symmetry – The Red Star of David
2D D{{sub|12}} symmetry — The Naval Jack of the Republic of China (White Sun)
2D D{{sub|24}} symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

Perimeter

Closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length.

Perimeter is the distance around a two dimensional shape, a measurement of the distance around something; the length of the boundary.
Perimeter of a rectangle.
If the diameter of a circle is 1, its circumference equals.
The more one cuts this shape, the lesser the area and the greater the perimeter. The convex hull remains the same.
The Neuf-Brisach fortification perimeter is complicated. The shortest path around it is along its convex hull.

The first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons.

Incircle and excircles of a triangle

[[Image:Incircle and Excircles.svg|right|thumb|300px|Incircle and excircles of a triangle.

The nine-point circle is tangent to the incircle and excircles

All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons.

Square

The area of a square is the product of the length of its sides.
Square at a given circumcircle
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r8 and no symmetry is labeled a1.
Crossed-square
3-simplex (3D)

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles).

Equilateral triangle

Equilateral triangle is a triangle in which all three sides have the same length.

The equilateral triangle tiling fills the plane.
A regular tetrahedron is made of four equilateral triangles.
Construction of equilateral triangle with compass and straightedge
Visual proof of Viviani's theorem

It is also a regular polygon, so it is also referred to as a regular triangle.

Hexagon

Six-sided polygon or 6-gon.

The six lines of reflection of a regular hexagon, with Dih6 or r12 symmetry, order 12.
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r12 and no symmetry is labeled a1.
Giant's Causeway closeup
Equilateral triangles on the sides of an arbitrary hexagon
A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D3d, [2+,6], (2*3), order 12.
The ideal crystalline structure of graphene is a hexagonal grid.
Assembled E-ELT mirror segments
A beehive honeycomb
The scutes of a turtle's carapace
Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet
Micrograph of a snowflake
Benzene, the simplest aromatic compound with hexagonal shape.
Hexagonal order of bubbles in a foam.
Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.
Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern
An aerial view of Fort Jefferson in Dry Tortugas National Park
The James Webb Space Telescope mirror is composed of 18 hexagonal segments.
Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France.
Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals
Hexagonal barn
The Hexagon, a hexagonal theatre in Reading, Berkshire
Władysław Gliński's hexagonal chess
Pavilion in the Taiwan Botanical Gardens
Hexagonal window

A regular hexagon has Schläfli symbol {6} and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.

Octagon

Octagon is an eight-sided polygon or 8-gon.

The diagonals of the green quadrilateral are equal in length and at right angles to each other
building a regular octagon by folding a sheet of paper
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Meccano octagon construction.
A regular skew octagon seen as edges of a square antiprism, symmetry D4d, [2+,8], (2*4), order 16.
The octagonal floor plan, Dome of the Rock.
Umbrellas often have an octagonal outline.
The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.
The street & block layout of Barcelona's Eixample district is based on non-regular octagons
Janggi uses octagonal pieces.
Japanese lottery machines often have octagonal shape.
Stop sign used in English-speaking countries, as well as in most European countries
An icon of a stop sign with a hand at the middle.
The trigrams of the Taoist bagua are often arranged octagonally
Famous octagonal gold cup from the Belitung shipwreck
Classes at Shimer College are traditionally held around octagonal tables
The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.
The movement of the analog stick(s) of the Nintendo 64 controller, the GameCube controller, the Wii Nunchuk and the Classic Controller is restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.
The truncated square tiling has 2 octagons around every vertex. {{CDD|node_1|4|node_1|4|node_1}}
An octagonal prism contains two octagonal faces. {{CDD|node_1|4|node_1|2|node_1}}
An octagonal antiprism contains two octagonal faces. {{CDD|node_h|8|node_h|2x|node_h}}
The truncated cuboctahedron contains 6 octagonal faces. {{CDD|node_1|4|node_1|3|node_1}}
The omnitruncated cubic honeycomb {{CDD|node_1|4|node_1|3|node_1|4|node_1}}

A regular octagon has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges.

Uniform polyhedron

Platonic solid: Tetrahedron
Uniform star polyhedron: Snub dodecadodecahedron
The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron
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Example forms from the cube and octahedron
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In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other).