Decagon

In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon. Decagons are interesting shapes that appear in many areas of math and design. The total sum of the interior angles of a simple decagon is 1440°, which helps mathematicians and scientists understand how these shapes behave. Because of their symmetry and balance, decagons are used in architecture, art, and even nature, where some flowers and viruses have structures that resemble this shape. Learning about decagons helps us explore the beauty and order found in geometry.

Regular decagon

A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.

Decagons often appear in tilings with (partial) 5-fold symmetry. The images show an Islamic geometric pattern (15th century), an illustration in Kepler's Harmonices Mundi (1619) and a Penrose tiling.

The regular decagon can be constructed with ruler and compass.

The golden ratio in decagon

The golden ratio plays an important role in creating a decagon. Whether you start with a circle around which the decagon is drawn or with the length of one side, the golden ratio helps determine the correct proportions.

In both methods, special lines and curves are used to divide segments in a way that follows the golden ratio, a number approximately equal to 1.618. This ratio appears in many natural patterns and shapes, making it useful in geometry and art.

Symmetry

The regular decagon has special patterns called symmetries that help us understand its shape. It belongs to a group called Dih₁₀, which has 20 different ways to move or flip the shape so it looks the same. There are also smaller symmetry groups, like Dih₅, Dih₂, Dih₁, Z₁₀, Z₅, Z₂, and Z₁.

These symmetries can be seen in different ways on the decagon, depending on whether the lines pass through corners or sides. Some irregular decagons, which are not perfectly shaped, can still have high symmetry. Two special irregular decagons are d10 and p10, which are related to each other in a special way called being duals.

Dissection

A regular decagon, which has ten sides, can be divided into ten special shapes called rhombi. This means you can split the decagon into smaller diamond-like pieces that fit together perfectly. This idea is part of a bigger pattern in geometry and connects to shapes found in higher-dimensional spaces.

Skew decagon

A skew decagon is a special kind of shape with 10 points and lines, but it doesn't lie flat on one surface. Instead, its points and lines twist through space. One special type is called a skew zig-zag decagon, where the points alternate between two flat surfaces.

A regular skew decagon has all its lines the same length and looks the same from any of its points. You can see this shape in certain 3D objects, like a pentagonal antiprism, and it helps describe more complex shapes in higher dimensions. It is called a Petrie polygon for many higher-dimensional figures.