Octagon

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, {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.

Properties

Assuming plane geometry, 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.

Regularity

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° (${\displaystyle \frac{3\pi }{4}}$ radians). The central angle is 45° (${\displaystyle \frac{\pi }{4}}$ radians).

Area

The area of a regular octagon of side length a is given by

${\displaystyle A=2\cot \frac{\pi }{8}{a}^2=2(1+\sqrt{2})a^2\approx 4.828a^2.}$

In terms of the circumradius R, the area is

${\displaystyle A=4\sin \frac{\pi }{4}{R}^2=2\sqrt{2}{R}^2\approx 2.828R^2.}$

In terms of the apothem r (see also inscribed figure), the area is

${\displaystyle A=8\tan \frac{\pi }{8}{r}^2=8(\sqrt{2}-1)r^2\approx 3.314r^2.}$

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

The area can also be expressed as

${\displaystyle A=S^2-a^2,}$

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 overlap with the four sides of the square) and then takes the corner triangles (these are 45–45–90 triangles) and places them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the length of a side a, the span S is

${\displaystyle S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a\approx 2.414a.}$

The span, then, is equal to the silver ratio times the side, a.

The area is then as above:

${\displaystyle A=((1+\sqrt{2})a{)}^2-a^2=2(1+\sqrt{2})a^2\approx 4.828a^2.}$

Expressed in terms of the span, the area is

${\displaystyle A=2(\sqrt{2}-1)S^2\approx 0.828S^2.}$

Another simple formula for the area is

${\displaystyle A=2aS.}$

More often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above,

${\displaystyle a\approx S/\mathrm{2.414.}}$

The two end lengths e on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being ${\displaystyle e=a/\sqrt{2},}$ may be calculated as

${\displaystyle e=(S-a)/2.}$

Circumradius and inradius

The circumradius of the regular octagon in terms of the side length a is

${\displaystyle R=\left(\frac{\sqrt{4+2\sqrt{2}}}{2}\right)a\approx 1.307a,}$

and the inradius is

${\displaystyle r=\left(\frac{1+\sqrt{2}}{2}\right)a\approx 1.207a.}$

(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

${\displaystyle r=R\cos \frac{\pi }{8}}$

Diagonality

The regular octagon, in terms of the side length a, has three different types of diagonals:

• Short diagonal;
• Medium diagonal (also called span or height), which is twice the length of the inradius;
• Long diagonal, which is twice the length of the circumradius.

The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:

• Short diagonal: ${\displaystyle a\sqrt{2+\sqrt{2}}}$ ;
• Medium diagonal: ${\displaystyle (1+\sqrt{2})a}$ ; (silver ratio times a)
• Long diagonal: ${\displaystyle a\sqrt{4+2\sqrt{2}}}$ .

Construction

A regular octagon at a given circumcircle may be constructed as follows:

1. Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
2. Draw another diameter GOC, perpendicular to AOE.
3. (Note in passing that A,C,E,G are vertices of a square).
4. Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
5. A,B,C,D,E,F,G,H are the vertices of the octagon.

A regular octagon can be constructed using a straightedge and a compass, as 8 = 2³, a power of two:

The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required.

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:

${\displaystyle \text{Area}=2a^2(\sqrt{2}+1)}$

for an octagon of side a.

Standard coordinates

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

• (±1, ±(1+√2))
• (±(1+√2), ±1).

Dissectibility

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list (sequence A006245 in the OEIS) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings.