Pentagon

Pentagon
Dimensionality	2
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Pentagonal dihedral symmetry (D5)
Coxeter–Dynkin diagram
Bowers acronym	peg
Subfacets
Vertices	5 points
Edges	5 line segments
Faces	1 pentagon
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'

A pentagon is a 2-dimensional polygon with five edges. The Bowers acronym for a pentagon is peg. It is called a cosmogon, hydrogon, or a rhodogon under the elemental naming scheme.

Structure and Sections

The regular pentagon has equal edges and each angle being 108 degrees.

Hypervolumes

• vertex count =$5$
• edge length =$5l$
• surface area =${\frac {\sqrt {5(5+2{\sqrt {5}})}}{4}}{l}^{2}$

Subfacets

• 1 null polytope (-1D)
• 5 points (0D)
• 5 line segments (1D)
• 1 pentagon (2D)

Radii

• Vertex radius:${\sqrt {\frac {5+{\sqrt {5}}}{10}}}l$
• Edge radius:${\frac {{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}{2}}{l}}$

Angles

• Dihedral angle: 108º

Vertex coordinates

The coordinates of a pentagon in the plane are:

• (0,√((10+2√5)/5)
• (±(1+√5)/2,√((5-√5)/10)
• (±1,-√((5+2√5)/5)

Related shapes

• Dual: Self-dual
• Vertex figure: Line segment, length φ =$\frac{1+\sqrt{5}}{2}$
• Conjugate: Pentagram

See Also

\(\{0\}\)	\(\{1\}\)	\(\{2\}\)	\(\{3\}\)	\(\{4\}\)	\(\{5\}\)	\(\{\frac{5}{2}\}\)	\(\{6\}\)	\(\{7\}\)	\(\{\frac{7}{2}\}\)	\(\{\frac{7}{3}\}\)	\(\{8\}\)	\(\{\frac{8}{3}\}\)	\(\{9\}\)	\(\{\frac{9}{2}\}\)	\(\{\frac{9}{4}\}\)	\(\{10\}\)	\(\{\frac{10}{3}\}\)	\(\{11\}\)	\(\{\frac{11}{2}\}\)	\(\{\frac{11}{3}\}\)	\(\{\frac{11}{4}\}\)	\(\{\frac{11}{5}\}\)	\(\{12\}\)	\(\{\frac{12}{5}\}\)	\(\{13\}\)	\(\{\frac{13}{2}\}\)	\(\{\frac{13}{3}\}\)	\(\{\frac{13}{4}\}\)	\(\{\frac{13}{5}\}\)	\(\{\frac{13}{6}\}\)	\(\{14\}\)	\(\{\frac{14}{3}\}\)	\(\{\frac{14}{5}\}\)	\(\{15\}\)	\(\{\frac{15}{2}\}\)	\(\{\frac{15}{4}\}\)	\(\{\frac{15}{7}\}\)	\(\{16\}\)	\(\{\frac{16}{3}\}\)	\(\{\frac{16}{5}\}\)	\(\{\frac{16}{7}\}\)	\(\{17\}\)	\(\{\frac{17}{2}\}\)	\(\{\frac{17}{3}\}\)	\(\{\frac{17}{4}\}\)	\(\{\frac{17}{5}\}\)	\(\{\frac{17}{6}\}\)	\(\{\frac{17}{7}\}\)	\(\{\frac{17}{8}\}\)	...	\(\{\infty\}\)	\(\{x\}\)	\(\{\frac{\pi i}{\lambda}\}\)
Zerogon	Monogon	Digon	Triangle	Square	Pentagon	Pentagram	Hexagon	Heptagon	Heptagram	Great heptagram	Octagon	Octagram	Enneagon	Enneagram	Great enneagram	Decagon	Decagram	Hendecagon	Small hendecagram	Hendecagram	Great hendecagram	Grand hendecagram	Dodecagon	Dodecagram	Tridecagon	Small tridecagram	Tridecagram	Medial tridecagram	Great tridecagram	Grand tridecagram	Tetradecagon	Tetradecagram	Great tetradecagram	Pentadecagon	Small pentadecagram	Pentadecagram	Great pentadecagram	Hexadecagon	Small hexadecagram	Hexadecagram	Great hexadecagram	Heptadecagon	Tiny heptadecagram	Small heptadecagram	Heptadecagram	Medial heptadecagram	Great heptadecagram	Giant heptadecagram	Grand heptadecagram	...	Apeirogon	Failed star polygon (\(x\)-gon)	Pseudogon (\(\frac{\pi i}{\lambda}\)-gon)

Regular $t_0 \{5\}$	Rectified $t_1 \{5\}$	Truncated $t_{0,1} \{5\}$
Pentagon	Pentagon	Decagon

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	Seventeen	...	Aleph null
Simplex \(\{3^{n-1}\}\)	Null polytope \()(\) \(\emptyset\)	Point \(()\) \(\mathbb{B}^0\)	Line segment \(\{\}\) \(\mathbb{B}^1\)	Triangle \(\{3\}\)	Tetrahedron \(\{3^2\}\)	Pentachoron \(\{3^3\}\)	Hexateron \(\{3^4\}\)	Heptapeton \(\{3^5\}\)	Octaexon \(\{3^6\}\)	Enneazetton \(\{3^7\}\)	Decayotton \(\{3^8\}\)	Hendecaxennon \(\{3^9\}\)	Dodecadakon \(\{3^{10}\}\)	Tridecahendon \(\{3^{11}\}\)	Tetradecadokon \(\{3^{12}\}\)	Pentadecatradakon \(\{3^{13}\}\)	Hexadecatedakon \(\{3^{14}\}\)	Heptadecapedakon \(\{3^{15}\}\)	Octadecapedakon \(\{3^{16}\}\)	...	Omegasimplex
Cross \(\{3^{n-2},4\}\)				Square \(\{4\}\)	Octahedron \(\{3, 4\}\)	Hexadecachoron \(\{3^2, 4\}\)	Pentacross \(\{3^3, 4\}\)	Hexacross \(\{3^4, 4\}\)	Heptacross \(\{3^5, 4\}\)	Octacross \(\{3^6, 4\}\)	Enneacross \(\{3^7, 4\}\)	Dekacross \(\{3^8, 4\}\)	Hendekacross \(\{3^9, 4\}\)	Dodekacross \(\{3^{10}, 4\}\)	Tridekacross \(\{3^{11}, 4\}\)	Tetradekacross \(\{3^{12}, 4\}\)	Pentadekacross \(\{3^{13}, 4\}\)	Hexadekacross \(\{3^{14}, 4\}\)	Heptadekacross \(\{3^{15}, 4\}\)	...	Omegacross
Hydrotopes \(\{3^{n-2}, 5\}\)				Pentagon \(\{5\}\)	Icosahedron \(\{3, 5\}\)	Hexacosichoron \(\{3^2, 5\}\)	Order-5 pentachoric tetracomb \(\{3^3, 5\}\)	Order-5 hexateric pentacomb \(\{3^4, 5\}\)	...
Hypercube \(\{4, 3^{n-2}\}\)				Square \(\{4\}\)	Cube \(\{4, 3\}\)	Tesseract \(\{4, 3^2\}\)	Penteract \(\{4, 3^3\}\)	Hexeract \(\{4, 3^4\}\)	Hepteract \(\{4, 3^5\}\)	Octeract \(\{4, 3^6\}\)	Enneract \(\{4, 3^7\}\)	Dekeract \(\{4, 3^8\}\)	Hendekeract \(\{4, 3^9\}\)	Dodekeract \(\{4, 3^{10}\}\)	Tridekeract \(\{4, 3^{11}\}\)	Tetradekeract \(\{4, 3^{12}\}\)	Pentadekeract \(\{4, 3^{13}\}\)	Hexadekeract \(\{4, 3^{14}\}\)	Heptadekeract \(\{4, 3^{15}\}\)	...	Omegeract
Cosmotopes \(\{5, 3^{n-2}\}\)				Pentagon \(\{5\}\)	Dodecahedron \(\{5, 3\}\)	Hecatonicosachoron \(\{5, 3^2\}\)	Order-3 hecatonicosachoric tetracomb \(\{5, 3^3\}\)	Order-3-3 hecatonicosachoric pentacomb \(\{5, 3^4\}\)	...
Hyperball \(\mathbb B^n\)				Disk \(\mathbb B^2\)	Ball \(\mathbb B^3\)	Gongol \(\mathbb B^4\)	Pentorb \(\mathbb B^5\)	Hexorb \(\mathbb B^6\)	Heptorb \(\mathbb B^7\)	Octorb \(\mathbb B^8\)	Enneorb \(\mathbb B^9\)	Dekorb \(\mathbb B^{10}\)	Hendekorb \(\mathbb B^{11}\)	Dodekorb \(\mathbb B^{12}\)	Tridekorb \(\mathbb B^{13}\)	Tetradekorb \(\mathbb B^{14}\)	Pentadekorb \(\mathbb B^{15}\)	Hexadekorb \(\mathbb B^{16}\)	Heptadekorb \(\mathbb B^{17}\)	...	Omegaball \(\mathbb B^{\aleph_0}\)