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) | $ 2\left(1+\sqrt2\right)a^2 $ $ \simeq4.828a^2 $ |
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 $ 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.

An octagon inset in a square.
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 $
Uses of octagons
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.
See also
External links
- How to find the area of an octagon
- Definition and properties of an octagon With interactive animation
- Weisstein, Eric W., "Octagon" from MathWorld.
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