Octagon

In geometry, an octagon is a shape with eight sides and eight angles. The word "octagon" comes from Ancient Greek. Octagons can be seen in buildings and sports fields.

A regular octagon has all sides and angles the same. It has a special symbol called the Schläfli symbol, written as {8}. You can think of a regular octagon as a square with its corners cut off neatly, making the shape of an octagon. This is called a truncated square.

If you cut off more corners from an octagon, you can make a shape with more sides, like a hexadecagon, which has sixteen sides. In three dimensions, shapes related to octagons can also exist, such as the rhombicuboctahedron.

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°.

8-cube projection	24 rhomb dissection
	Regular	Isotoxal

Skew

A skew octagon is a special eight-sided shape. In a skew octagon, the corners and sides are not all on the same flat surface. This makes it tricky to know what is inside the shape.

A regular skew octagon has all its sides the same length. You can see this shape in some 3D objects, like the edges of a square antiprism. It is also connected to more complicated shapes in higher dimensions.

Symmetry

The regular octagon has a special kind of balance called Dih₈ symmetry. This means you can flip and turn it in many ways, and it will still look 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 look the same, but each has half the symmetry of a regular octagon.

Symmetry
	The eleven symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.

Example octagons by symmetry
r16
d8	g8	p8
d4	g4	p4
d2	g2	p2
a1

Use

The octagonal shape is used a lot in buildings and design. Famous places 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 have octagonal shapes.

Octagons are also found in everyday items. Umbrellas often have an octagonal outline, and stop signs in English-speaking countries and most European countries are octagonal. Games like Janggi use octagonal pieces, and some lottery machines in Japan have an octagonal shape. Also, 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 has octagons at each point. An octagonal prism has two octagon-shaped ends, just like an octagonal antiprism. The truncated cuboctahedron has six octagon-shaped faces. The omnitruncated cubic honeycomb includes octagons in its design.

Related polytopes

The octagon, when seen as a shortened square, begins a group of shortened hypercubes. It is also the first in a group of enlarged hypercubes, starting with an expanded square.