Icosagon

In geometry, an icosagon or 20-gon is a special shape that has twenty sides. It belongs to a group of shapes called polygons, which are flat figures made by joining straight lines end-to-end. One important fact about any icosagon is that the total measurement of all its inside angles adds up to 3240 degrees. This shape can be found in many areas, from art and architecture to math problems and puzzles, showing how useful geometry is in understanding the world around us.

Regular icosagon

A regular icosagon is a special shape with twenty straight sides that are all the same length. Each inside corner, or angle, of this shape measures 162 degrees.

The space, or area, inside a regular icosagon can be calculated using a special math formula. If you know the length of one side, the area is about 31.57 times the side length multiplied by itself. This shape takes up about 98.36% of the space inside a circle that just fits around it.

Uses

The Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section.

The Globe, the outdoor theater used by William Shakespeare's acting company, was discovered to have been built on an icosagonal foundation when a partial excavation was done in 1989.

As a golygonal path, the swastika is considered to be an irregular icosagon.

A regular square, pentagon, and icosagon can completely fill a plane vertex.

Construction

As 20 = 2² × 5, regular icosagon is constructible using a compass and straightedge, or by an edge-bisection of a regular decagon, or a twice-bisected regular pentagon:

The golden ratio in an icosagon

In geometry, a special shape called an icosagon has twenty sides. When building this shape with a certain side length, a circle drawn around a point can split a part of the shape in a special way. This special split is known as the golden ratio, which is approximately 1.618. This means that the lengths of certain parts of the shape relate to each other in this special number way.

Symmetry

An icosagon is a shape with twenty sides. When it is perfectly regular, it has a special kind of balance called Dih₂₀ symmetry, which means it can be flipped and turned in many ways and still look the same. This symmetry can be broken down into smaller patterns, including five types that allow flipping and six types that only allow turning.

These smaller patterns create sixteen different ways the shape can look while keeping some balance. Each pattern is given a special name, like r40 for the full balance or a1 for no balance at all. Some patterns match up with corners, some with sides, and others with both.

Even when the shape is not perfect, it can still keep some of this balance. Two special imperfect shapes keep half the balance of the perfect shape. One changes the lengths of its sides, while the other changes the angles between its sides.

Dissection

An icosagon, which is a shape with twenty sides, can be split into smaller shapes called parallelograms. For a regular icosagon, these smaller shapes are special types of parallelograms called rhombi, and there are also some squares. Specifically, it can be divided into 5 squares and 4 groups of 10 rhombi, making a total of 45 smaller shapes. This way of splitting the shape comes from a special view of a ten-dimensional cube.

Related polygons

An icosagram is a 20-sided star polygon, shown by the symbol {20/n}. There are three regular shapes for this, called {20/3}, {20/7}, and {20/9}. There are also five special star shapes that use the same points: 2{10}, 4{5}, 5{4}, 2{10/3}, 4{5/2}, and 10{2}.

Changing a regular decagon or decagram in certain ways can make special icosagram shapes that have evenly spaced points and two different lengths of lines.

A regular icosagram, {20/9}, looks like a changed decagon, t{10/9}={20/9}. In the same way, a decagram, {10/3} can be changed to t{10/7}={20/7}, and a simple change of a decagram gives t{10/3}={20/3}.

Petrie polygons

The regular icosagon is the Petrie polygon for several higher-dimensional shapes. You can see it in special views called Coxeter planes.

It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell, great icosahedral 120-cell, and great grand 120-cell.