1,183 Pages

Regular octagon A regular octagon
Edges and vertices 8
Schläfli symbols {8}
t{4}
Coxeter–Dynkin diagrams      Symmetry group Dihedral (D8)
Area
(with a=edge length)  Internal angle
(degrees)
135°

In geometry, an octagon is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

## Regular octagons A regular octagon is constructible with compass and straightedge. To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 through 10.

A regular octagon is always an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080°. The area of a regular octagon of side length is given by In terms of , (circumradius) the area is In terms of , (inradius) the area is Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.

The area can also be derived as folllows: where is the span of the octagon, or the second shortest diagonal; and is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45-45-90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the span , the length of a side is:  The area, is then as above: ## Derived figures

### Petrie polygons

The octagon is the Petrie polygon for these 12 higher-dimensional uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes.