Decagon

A decagon is a special shape.

It has ten sides. It also has ten corners.

A regular decagon is very neat. All its sides are the same length. Every corner looks the same, too.

You can make this shape with tools. You can use a ruler and a compass. You can even make it from a five-sided shape.

This shape is fun to study. It can be split into ten smaller diamond shapes.

A decagon is a shape with ten sides. It also has ten corners, which we call angles.

In a regular decagon, everything is very even. Every side has the same length. Every corner also looks the same. Each corner has an angle of 144 degrees. If you add all the corners together, you get 1440 degrees.

You can make a regular decagon using simple tools. You can use a ruler and a compass. You can even make one by using a five-sided shape, called a pentagon. One way is to draw a pentagon inside a circle. Then, draw lines from the corners through the center. Where those lines hit the circle, you find the new corners.

This shape is also full of patterns. You can split a regular decagon into ten smaller diamond shapes. These shapes are called rhombi.

Sometimes, decagons do not lie flat. A skew decagon zig-zags through space. It does not stay on one flat surface. These shapes can be seen in many 3D objects.

A decagon is a special shape in geometry. The name comes from Greek words that mean "ten angles."

This shape is a polygon with ten sides. A polygon is just a flat shape made of straight lines. You can have many different kinds of decagons. Some might have sides of different lengths. Others might have corners that look different. However, a regular decagon is very balanced. In a regular decagon, every side is the same length. Every internal angle is also exactly 144 degrees.

If you add all ten angles together, the total sum is 1440 degrees.

There are many ways to build a regular decagon. You can use a ruler and a compass to draw one. This is possible because ten is a power of two times a Fermat prime.

One clever way involves using a five-sided shape called a pentagon. First, you draw a pentagon inside a circle. Then, you draw lines from each corner through the center of the circle. Where those lines touch the edge of the circle, you find the new corners. These new points, along with the original ones, make the ten corners of a decagon. You can also think of it as a truncated pentagon. This means you take a pentagon and cut off its corners in a specific way.

Mathematics also looks at the patterns and symmetry of the shape. A regular decagon has a lot of symmetry. This means you can flip or turn it and it still looks the same.

It has 20 different ways to show symmetry. Some of these ways happen by reflecting the shape across lines. These lines can pass through the corners or through the middle of the sides. There are also ways to rotate the shape around its center. These are called cyclic symmetries. Mathematicians use special names like Dih10 to describe these patterns. Even irregular decagons can have some symmetry if they follow certain rules.

Decagons can also be broken down into smaller pieces. This is called a dissection.

For a regular decagon, you can divide it into ten smaller diamond shapes. These diamonds are called rhombi.

This works because a decagon is a type of zonogon. A zonogon is a shape where opposite sides are parallel and equal. This pattern of ten rhombi is very special. It is related to how shapes look when they are projected from higher dimensions. For example, it can be seen in a projection of a 5-cube. This connects a simple flat shape to much larger ideas in math.

Sometimes, decagons do not stay flat on a piece of paper. These are called skew decagons.

A skew decagon zig-zags through space instead of lying on one plane. Its edges and corners move up and down between two different levels. You can find these zig-zag shapes in 3D objects like a pentagonal antiprism.

They are also found in shapes called dodecahedrons and icosahedrons. In higher math, these zig-zag paths are called Petrie polygons. They show up when we look at very complex shapes from a certain angle. Even a shape with ten sides can lead to very big discoveries.

A decagon is a polygon characterized by having ten sides and ten angles. The name is derived from the Greek words "deka," meaning ten, and "gonia," meaning angle.

In geometry, a simple decagon has an interior angle sum of 1440 degrees. While decagons can be irregular, a regular decagon is defined by total equality. Every side in a regular decagon is the same length. Additionally, every internal angle measures exactly 144 degrees. This specific shape can be represented by the Schläfli symbol {10}.

Constructing a regular decagon is a classic geometric challenge. It is possible to build one using only a compass and a straightedge. This feasibility exists because the number ten is a power of two multiplied by a Fermat prime.

One method involves the use of a regular pentagon. You can construct a pentagon inside a circle first. Then, you draw lines from each vertex through the center of the circle. The points where these lines intersect the circle become the new vertices. By joining these new points with the original pentagon vertices, you form a decagon. Another way to view it is as a truncated pentagon, denoted as t{5}.

Mathematical construction often involves the golden ratio. This ratio is a fundamental constant that helps divide line segments through exterior division. In a decagon constructed within a given circumcircle, a circular arc produces a segment that corresponds to this ratio.

Similarly, when constructing a decagon from a given side length, the golden ratio appears in the division of segments. This relationship between the decagon and the golden ratio is a deep connection in geometry. It highlights how simple shapes are often tied to universal mathematical constants.

Symmetry describes how a shape can be moved and still look identical. A regular decagon possesses Dih10 symmetry, which has an order of 20.

This includes ten distinct symmetries involving reflections and rotations. The reflection lines, or mirrors, can pass through the vertices or through the edges. Mathematicians like John Conway use specific labels to describe these patterns. For example, reflections through vertices are labeled "d" for diagonal, while reflections through edges are labeled "p" for perpendiculars. The shape also features cyclic group symmetries, such as Z10, Z5, Z2, and Z1.

Even irregular decagons can exhibit high levels of symmetry. An isogonal decagon, labeled d10, is constructed by five mirrors. This shape can alternate between long and short edges. Conversely, an isotoxal decagon, labeled p10, has equal edge lengths but alternating internal angles. These two types of irregular decagons are considered duals of each other. They each possess exactly half the symmetry order of a regular decagon. This demonstrates how varying one property can create new, organized structures.

Decagons can also be broken down into smaller, simpler shapes through a process called dissection.

A regular decagon is a type of zonogon. A zonogon is a polygon where opposite sides are parallel and equal in length. Because it is a zonogon with an even number of sides, it can be dissected into rhombi. For a decagon, this means it can be divided into exactly 10 rhombi.

This specific decomposition is related to the projection of a 5-cube. It also relates to the faces of a rhombic triacontahedron.

Finally, decagons do not always have to lie flat on a single plane. A skew decagon is a polygon with ten sides that zig-zags through three-dimensional space.

These shapes are vertex-transitive and have equal edge lengths. They can be seen in the edges of a pentagonal antiprism or a pentagrammic antiprism. In higher-dimensional mathematics, these are known as Petrie polygons. They appear as the perimeter of projections of complex shapes like the dodecahedron or the icosahedron.

This shows that a ten-sided shape connects simple geometry to the study of complex, high-dimensional polytopes.