Octagon

Octagon
Dimensionality	2
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Octagonal dihedral symmetry (D8)
Bowers acronym	oc
Subfacets
Vertices	8 points
Edges	8 line segments
Faces	1 octagon
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'

An octagon is a 2-dimensional polygon with eight edges. The Bowers acronym for an octagon is oc.

Symbols

Symbols for the octagon include:

• x8o (regular)
• x4x (ditetragon)
• ko4ok&#zx (tetrambus)
• xwwx&#zx (edge first)
• xw wx&#zx (rectangle symmetry)

Structure and Sections

The regular octagon has each angle at 135 degrees. It can be seen as a truncated square.

Hypervolumes

• vertex count = $8$
• edge length = $8l$
• surface area = $2\left( 1+\sqrt{2} \right) {l}^{2}$

Subfacets

• 8 points (0D)
• 8 line segments (1D)
• 1 octagon (2D)

Radii

• Vertex radius: $\sqrt{\frac{2+\sqrt{2}}{2}}l$
• edge radius; $\frac{1+\sqrt{2}}{2}l$

Angles

• Vertex angle: 135º

Vertex coordinates

The coordinates of a regular octagon are all permutations and sign changes of (1,1+√2).

Related shapes

• Dual: Self-dual
• Vertex figure and short diagonal: Line segment, length $\sqrt{2+\sqrt{2}}$
• Medium diagonal: Length $1+\sqrt {2}$
• Long diagonal: Length $\sqrt{4+2\sqrt{2}}$

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 $t_0 \{8\}$	Rectified $t_1 \{8\}$	Truncated $t_{0,1} \{8\}$
Octagon	Octagon	Hexadecagon

Regular $t_0 \{4\}$	Rectified $t_1 \{4\}$	Truncated $t_{0,1} \{4\}$
Square	Square	Octagon

Regular $t_0 \{\frac{8}{7} \}$	Rectified $t_1 \{\frac{8}{7} \}$	Truncated $t_{0,1} \{\frac{8}{7} \}$
Octagon	Octagon	Great hexadecagram