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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 $a$ is given by

$A=2\cot\left(\frac{\pi}{8}\right)a^2=2\left(1+\sqrt2\right)a^2\simeq4.828a^2$

In terms of $R$ , (circumradius) the area is

$A=4\sin\left(\frac{\pi}{4}\right)R^2=2\sqrt2R^2\simeq2.828R^2$

In terms of $r$ , (inradius) the area is

$A=8\tan\left(\frac{\pi}{8}\right)r^2=8\left(\sqrt2-1\right)r^2\simeq3.313r^2$

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.

The area can also be derived as folllows:

$A=S^2-a^2$

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. 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 $S$ , the length of a side $a$ is:

$S=\frac{a}{\sqrt2}+a+\frac{a}{\sqrt2}=\left(1+\sqrt2\right)a$

$S=2.414a$

The area, is then as above:

$A=\bigl((1+\sqrt2)a\bigr)^2-a^2=2\left(1+\sqrt2\right)a^2$

## 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.