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

A regular octagon has Schläfli symbol {8}[1] and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges.

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).[2]: Prop.

10 A regular octagon is a closed figure with sides of the same length and internal angles of the same size.

The area can also be expressed as where S is the span of the octagon, or the second-shortest diagonal; and a is the length of one of the sides, or bases.

may be calculated as The circumradius of the regular octagon in terms of the side length a is[3] and the inradius is (that is one-half the silver ratio times the side, a, or one-half the span, S) The inradius can be calculated from the circumradius as The regular octagon, in terms of the side length a, has three different types of diagonals: The formula for each of them follows from the basic principles of geometry.

Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices.

Its area can thus be computed as the sum of eight isosceles triangles, leading to the result: for an octagon of side a.

[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi.

The list (sequence A006245 in the OEIS) defines the number of solutions as eight, by the eight orientations of this one dissection.

In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D4d, [2+,8] symmetry, order 16.

The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan.

Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in the Intelsat Headquarters of Washington or Callam Offices in Canberra.