Decagon

In geometry, a decagon is a shape with ten sides. It comes from the Greek words for "ten" and "angles". Decagons are special shapes found in math, design, and nature.

The inside angles of a decagon add up to 1440°. This helps us learn how these shapes work.

Because decagons are balanced and symmetrical, they are used in buildings, art, and even in nature. Some flowers and tiny living things have shapes like decagons. Studying decagons helps us see the order and beauty in geometry.

Regular decagon

A regular decagon has all sides the same length. Each internal angle is always 144°. Its Schläfli symbol is {10}. It can also be made by changing the shape of a pentagon, written as t{5}.

Decagons often appear in designs with a special kind of balance called 5-fold symmetry. The images show an Islamic geometric pattern from the 15th century, an illustration from Kepler's Harmonices Mundi from 1619, and a Penrose tiling.

The regular decagon can be drawn using just a ruler and compass.

The golden ratio in decagon

The golden ratio helps create a decagon. You can start with a circle or the length of one side. The golden ratio makes sure the shapes are the right size.

Special lines and curves divide parts to follow the golden ratio. This number is about 1.618. It appears in many natural patterns and shapes. This makes it useful in geometry and art.

Symmetry

The regular decagon has special patterns called symmetries. These symmetries help us understand its shape. It belongs to a group called Dih₁₀. This group 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. Some irregular decagons, which are not perfectly shaped, can still have high symmetry. Two special irregular decagons are d10 and p10. These two are related to each other in a special way called being duals.

Dissection

A regular decagon has ten sides. It can be split into ten special shapes called rhombi. These shapes look like diamonds and fit together perfectly. This idea is part of a bigger pattern in geometry. It also connects to shapes found in higher-dimensional spaces.

Skew decagon

A skew decagon is a special shape with 10 points and lines, but it does not lie flat. Its points and lines twist through space. One type is called a skew zig-zag decagon, where the points switch between two flat surfaces.

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