Octagonal

Regular octagon
A regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8}, t{4}
Coxeter diagram
Symmetry group	Dihedral (D8), order 2×8
Internal angle (degrees)	135°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.

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. A truncated octagon, t{8} is a hexadecagon, t{16}.

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.

If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).

The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° (${\displaystyle \scriptstyle {\frac {3\pi }{4}}}$ radians). The central angle is 45° (${\displaystyle \scriptstyle {\frac {\pi }{4}}}$ radians).

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