Octagon

Contents

1. Overview
2. Etymology
3. Cultural Impact

You want an article on octagons, do you? Fine. Don’t expect me to hold your hand through the geometric intricacies. Consider this a dissection, not a tutorial.

An octagon, derived from the Ancient Greek term “oktágōnon” (ὀκτάγωνον), meaning “eight angles,” is fundamentally a polygon characterized by its eight sides and, consequently, eight vertices. The term “octagonal” often redirects here, indicating the inherent association with this eight-sided figure. For other, less significant uses, one might consult Octagon (disambiguation) and Octagonal (disambiguation) .

Regular Octagon

The epitome of an octagon is its regular form. This is a regular polygon possessing eight equal edges and eight equal vertices . Its geometric designation is the Schläfli symbol {8}. It can also be conceptualized as a truncated square , denoted as t{4}, which implies an alternating pattern of two distinct edge types. Conversely, a truncated octagon, t{8}, results in a hexadecagon , or {16}. In a more abstract, three-dimensional sense, the rhombicuboctahedron can be seen as a 3D analog, with its triangular faces mirroring the replaced edges of a conceptual octagon derived from a truncated square.

Properties

The sum of the internal angles within any octagon, regardless of its regularity, amounts to 1080°. This is consistent with the general rule for polygons, where the external angles invariably sum to 360°.

A rather intriguing property emerges when squares are constructed either entirely on the interior or exterior of an octagon’s sides. The midpoints of the segments connecting the centers of opposite squares then form a quadrilateral exhibiting both equidiagonal and orthodiagonal characteristics. This means its diagonals are not only of equal length but also intersect at a right angle. This geometric observation is detailed in Prop. 9 .

Furthermore, if one considers the midpoint polygon of a reference octagon—a polygon whose vertices lie at the midpoints of the reference octagon’s sides—and subsequently constructs squares internally or externally on these midpoint sides, the midpoints of the connecting segments between opposite squares will, astonishingly, form the vertices of a square. This is elaborated in Prop. 10 .

Regularity

A regular octagon is a closed figure defined by sides of uniform length and internal angles of identical measure. It possesses eight lines of reflective symmetry and exhibits rotational symmetry of the eighth order. As previously mentioned, its Schläfli symbol is {8}.

The internal angle at each vertex of a regular octagon measures precisely 135 degrees, which is equivalent to 3π/4 radians . The central angle , the angle subtended by each side at the center of the octagon, is 45 degrees, or π/4 radians.

Area

The area of a regular octagon with a side length denoted by ‘a’ is expressed by the formula:

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

When considering the circumradius R, the area calculation transforms to:

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

In relation to the apothem r (also relevant to inscribed figures ), the area formula becomes:

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

It’s a curious mathematical coincidence that these last two coefficients, when applied to the unit circle, bracket the value of pi .

The calculation of a regular octagon’s area can be visualized as that of a truncated square .

Alternatively, the area can be derived using the formula $A = S^2 - a^2$, where ‘S’ represents the “span” of the octagon—its second-shortest diagonal—and ‘a’ is the length of one of its sides. This can be readily understood by imagining a square encompassing the octagon, such that four of the octagon’s sides align with the square’s sides. The four corner triangles, which are 45–45–90 triangles , can then be detached and reconfigured, with their right-angle vertices pointing inward, to form a smaller square. The sides of this inner square would each correspond to the base length ‘a’ of the octagon.

Given a side length ‘a’, the span ‘S’ is calculated as:

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

This means the span is precisely the silver ratio multiplied by the side length.

Substituting this into the area formula yields:

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

Expressed in terms of the span ‘S’, the area simplifies to:

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

A more direct formula for the area, when the span is known, is $A = 2aS$.

Often, the span ‘S’ is the known quantity, and the side length ‘a’ must be determined, such as when cutting an octagon from a square piece of material. In such cases, ‘a’ can be approximated by $a \approx S / 2.414$. The lengths of the two segments, ’e’, at each end of a side (the legs of the green triangles in the diagram), are $e = a / \sqrt{2}$, and can also be calculated as $e = (S - a) / 2$.

Circumradius and Inradius

For a regular octagon with side length ‘a’, the circumradius is given by:

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

The inradius , conversely, is:

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

This inradius is precisely half the silver ratio multiplied by the side length ‘a’, or half the span ‘S’. The inradius can also be derived from the circumradius using the formula $r = R \cos(\frac{\pi}{8})$.

Diagonality

A regular octagon, defined by its side length ‘a’, possesses three distinct types of diagonals :

Short diagonal: $a\sqrt{2+\sqrt{2}}$
Medium diagonal (span or height): $(1+\sqrt{2})a$ (this is the silver ratio multiplied by ‘a’)
Long diagonal: $a\sqrt{4+2\sqrt{2}}$ (this is twice the circumradius)

These formulas are derived from fundamental geometric principles.

Construction

A regular octagon can be constructed with respect to a given circumcircle. The process involves drawing a circle, then two perpendicular diameters, say AOE and GOC, where O is the center. The points A, C, E, and G form the vertices of a square. Subsequently, bisecting the right angles GOA and EOG yields two more diameters, HOD and FOB. The points A, B, C, D, E, F, G, and H then represent the vertices of the regular octagon.

The construction of a regular octagon from a given side length follows a similar procedure to that of a hexadecagon at a given side length .

The regular octagon is constructible using a straightedge and compass because 8 is a power of two ($8 = 2^3$).

It can also be constructed using Meccano bars, requiring twelve bars of size 4, three of size 5, and two of size 6.

Each side of a regular octagon subtends an angle of 45° at the center. Therefore, its area can be calculated as the sum of eight isosceles triangles, leading to the formula:

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

Standard Coordinates

For a regular octagon centered at the origin with a side length of 2, the vertices can be represented by the coordinates:

(±1, ±(1+√2))
(±(1+√2), ±1)

Dissectibility

Coxeter posited that any zonogon —a polygon with an even number of sides where opposite sides are parallel and equal in length—can be dissected into $m(m-1)/2$ parallelograms. This holds true for regular polygons with an even number of sides, where these parallelograms are specifically rhombi. For the regular octagon, $m=4$, and it can be dissected into 6 rhombi. This decomposition is visually represented in projections of the tesseract . The number of distinct solutions for this dissection is eight, corresponding to the eight possible orientations. These rhombi and squares are fundamental to Ammann–Beenker tilings .

Skew Octagons

A skew octagon is a skew polygon comprising eight vertices and edges that do not lie on the same plane. The concept of an “interior” for such an octagon is not standard. A skew zig-zag octagon is characterized by vertices that alternate between two parallel planes.

A regular skew octagon exhibits vertex-transitivity and has edges of equal length. In three dimensions, it manifests as a zig-zag skew octagon and can be observed in the vertices and edges of a square antiprism , possessing D_4d symmetry, denoted as [2+,8], (2*4), with an order of 16.

Petrie Polygons

The regular skew octagon serves as the Petrie polygon for various higher-dimensional regular and uniform polytopes . These are often depicted in skew orthogonal projections within specific Coxeter planes . Examples include the 7-simplex , 5-demicube , 16-cell , and tesseract .

Symmetry

The regular octagon exhibits Dih₈ symmetry, with an order of 16. It encompasses three dihedral subgroups: Dih₄, Dih₂, and Dih₁, along with four cyclic subgroups : Z₈, Z₄, Z₂, and Z₁ (the trivial group, implying no symmetry).

There are eleven distinct symmetries associated with the regular octagon. John Conway denotes the full symmetry as r16. Dihedral symmetries are categorized based on whether their reflection lines pass through vertices (d for diagonal) or edges (p for perpendiculars). Cyclic symmetries are marked with ‘g’ to indicate their central gyration orders. The complete symmetry of the regular form is r16, while a1 signifies no symmetry.

The most commonly encountered high-symmetry octagons are p8, an isogonal octagon formed by four alternating mirrors creating long and short edges, and d8, an isotoxal octagon with equal edge lengths but vertices featuring alternating internal angles. These two forms are duals of each other and possess half the symmetry order of the regular octagon. Each subgroup symmetry allows for variations in irregular octagonal forms, with the g8 subgroup being unique in having no degrees of freedom, essentially representing directed edges.

Use

The octagonal form is frequently employed in architectural design. Notable examples include the Dome of the Rock in Jerusalem , with its distinctive octagonal plan, and the Tower of the Winds in Athens. This shape also appears in ecclesiastical architecture, seen in structures like St. George’s Cathedral, Addis Ababa , the Basilica of San Vitale in Ravenna, Castel del Monte in Apulia, the Florence Baptistery , the Zum Friedefürsten Church in Germany, and numerous octagonal churches in Norway . The central space of Aachen Cathedral , the Palatine Chapel built during the Carolingian era, features a regular octagonal floor plan. Lesser architectural elements, such as the octagonal apse of Nidaros Cathedral , also incorporate this shape.

Architects like John Andrews have utilized octagonal layouts to functionally separate office areas from building services, as demonstrated in the Intelsat Headquarters in Washington and the Callam Offices in Canberra.

Further applications include:

Umbrellas frequently adopt an octagonal outline.
The renowned Bukhara rug design incorporates an octagonal “elephant’s foot” motif.
The street and block layout of Barcelona ’s Eixample district is based on non-regular octagons.
Janggi (Korean chess) utilizes octagonal pieces.
Japanese lottery machines are often octagonal.
A Stop sign , universally recognized, is octagonal.
The trigrams of the Taoist bagua are commonly arranged in an octagonal pattern.
A notable octagonal gold cup was recovered from the Belitung shipwreck .
Classes at Shimer College are traditionally conducted around octagonal tables.
The Labyrinth of the Reims Cathedral possesses a quasi-octagonal form.
The range of motion for the analog stick on controllers such as the Nintendo 64 controller , GameCube controller , Wii Nunchuk , and Classic Controller is constrained by an octagonal frame, aiding users in aiming in cardinal directions while allowing for circular movement.
Chairs from A la Ronde feature octagonal seats and backs, designed as a set of eight.

Derived Figures

The truncated square tiling features two octagons meeting at each vertex.
An octagonal prism has two octagonal faces.
An octagonal antiprism also contains two octagonal faces.
The truncated cuboctahedron is characterized by six octagonal faces.
The omnitruncated cubic honeycomb incorporates octagonal elements.

The octagon, as a truncated square , stands as the initial member in a sequence of truncated hypercubes .

The octagon also serves as the first element in a sequence of expanded hypercubes, when considered as an expanded square.