Icosagon

In geometry, an icosagon or 20-gon is a shape with twenty sides. It is a type of polygon, which are flat shapes made by connecting straight lines. One key fact about any icosagon is that the sum of all its inside angles equals 3240 degrees. This shape appears in art, architecture, and math puzzles, showing how useful geometry can be.

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 found with a math formula. If you know the length of one side, the area is about 31.57 times the side length times 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 found to have been built on an icosagonal foundation.

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

Construction

As 20 = 2² × 5, a 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 shape called an icosagon has twenty sides. When you draw a circle around a point in this shape, it can split a part of the shape in a special way. This special split is called the golden ratio, which is about 1.618. This means that some parts of the shape relate to each other using this special number.

Symmetry

An icosagon is a shape with twenty sides. When it is perfectly regular, it has a special kind of balance called Dih₂₀ symmetry. This means you can flip and turn the shape many times, and it will still look the same.

These smaller patterns create sixteen different ways the shape can look while keeping some balance. Each pattern has a special name, like r40 for 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 is a shape with twenty sides. It can be split into smaller shapes called parallelograms. For a regular icosagon, these are special shapes called rhombi, and there are also some squares. It can be divided into 5 squares and 4 groups of 10 rhombi, making a total of 45 smaller shapes. This 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: {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.

A regular icosagram, {20/9}, looks like a changed decagon. In the same way, a decagram, {10/3} can be changed.

Petrie polygons

The regular icosagon is the Petrie polygon for some 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.