Octagon

In geometry, an octagon is a special shape with eight sides and eight angles. The word "octagon" comes from Ancient Greek, where "ὀκτάγωνον" (oktágōnon) means "eight angles." Octagons can be found in many places, from buildings to sports fields.

A regular octagon is when all the sides and angles are equal. It has a special symbol called the Schläfli symbol, written as {8}. You can imagine a regular octagon as a square that has its corners cut off in a very neat way, creating the shape of an octagon. This process is known as a truncated square.

If you keep cutting off more corners from an octagon, you can create a shape with even more sides, like a hexadecagon, which has sixteen sides. In three dimensions, shapes related to octagons can also exist, such as the rhombicuboctahedron, a fascinating solid with many different faces.

Properties

The sum of all internal angles of any octagon is 1080°, while the total of all external angles is always 360°, just like with every polygon.

A regular octagon has all sides and angles equal. It features eight lines of reflective symmetry and can be rotated into eight matching positions. Each internal angle in a regular octagon measures exactly 135°.

Skew

A skew octagon is a special kind of eight-sided shape where the corners and sides are not all on the same flat surface. This means you can't always clearly say what's inside the shape.

A regular skew octagon has all its sides the same length and can be found in certain 3D shapes, like the edges of a square antiprism. It is also linked to more complex geometric shapes in higher dimensions.

Symmetry

The regular octagon has a special kind of balance called Dih₈ symmetry, which means it can be flipped and turned in many ways while still looking the same. There are eleven different ways to see this symmetry in a regular octagon.

Two common types of octagons with high symmetry are p8 and d8. The p8 octagon has alternating long and short edges, while the d8 octagon has equal edges but alternating angles. These two shapes are special because they are duplicates of each other, but each has half the symmetry of a regular octagon.

Use

The octagonal shape is commonly used in architecture and design. Famous buildings like the Dome of the Rock and the Tower of the Winds in Athens have octagonal designs. Many churches, such as St. George's Cathedral, Addis Ababa and the Florence Baptistery, also feature octagonal plans.

Octagons appear in everyday objects too. Umbrellas often have an octagonal outline, and stop signs in English-speaking and most European countries are octagonal. Games like Janggi use octagonal pieces, and some lottery machines in Japan have an octagonal shape. Additionally, the movement of analog sticks on classic video game controllers like the Nintendo 64 controller is bounded by an octagonal frame.

Derived figures

The truncated square tiling features octagons at each vertex. An octagonal prism has two octagonal faces, as does an octagonal antiprism. The truncated cuboctahedron includes six octagonal faces. The omnitruncated cubic honeycomb also incorporates octagons in its structure.

Related polytopes

The octagon, when thought of as a truncated square, starts a series of truncated hypercubes. It is also the first in a series of expanded hypercubes, beginning with an expanded square.