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Icosahedron

The ancient Greeks gave the polyhedron a name according to the number of faces. “Ikosi” means twenty, “hedra” means a face (Icosahedron is a solid with twenty faces).

Therefore, the question “What is an icosahedron?” Can be given the following definition: “An icosahedron is a geometric solid of twenty faces, each of which is a right triangle.”

The polyhedron belongs to regular polyhedra and is one of the five Platonic solids.
The icosahedron has the following characteristics:

The face type is a regular triangle;
The number of sides at the verge 3;
The total number of faces is 20;
The number of edges adjacent to the top is 5;
The total number of vertices is 12;
The total number of edges is 30.

 

The regular icosahedron is composed of twenty equilateral triangles. Each vertex of an icosahedron is a vertex of five triangles. Therefore, the sum of flat angles at each vertex is 300° (60° * 5).
The icosahedron has a center of symmetry - the center of the icosahedron has 15 axes of symmetry and 15 planes of symmetry.

Mathematical characteristics of icosahedron

icosahedron placed in a sphereAn icosahedron can be placed in a sphere (inscribed), so that each of its vertices will touch the inner wall of the sphere.

The radius of the described sphere of the icosahedron

 

The radius of the described sphere of the icosahedron     where "a" - is the side length.

The sphere inscribed inside the icosahedron

The sphere can be inscribed inside the icosahedron.

The radius of the icosahedron inscribed sphere

 

The radius of the icosahedron inscribed sphere

The surface area of the icosahedronThe surface area of the icosahedron

The surface area of the icosahedron can be represented as a surface area shape net. The surface area can be defined as the area of one of the sides of the icosahedron (this is the area of a regular triangle) multiplied by 20. Or use the formula:

formula The surface area of the icosahedron

The volume of the icosahedron

The volume of the icosahedron is determined by the following formula:

formula The volume of the icosahedron

Icosahedron shape nets

You can make an icosahedron by yourself. Paper or cardboard is the most suitable option. For assembly, you will need a paper net—a single sheet with lines for all the folds.

Plato

Choose a color for your polyhedron.

The ancient Greek philosopher, Plato, associated the icosahedron with the "earth" element - water. Therefore, to build a model of this regular polyhedron, we chose blue.

 icosahedron net

The figure shows an icosahedron net:

Note that this is not the only option for a icosahedron net.

To build a model, you can download a net in pdf format and print it on an A4 sheet:
- if you print on a color printer - color net.
- if you intend to use colored cardboard for the assembly - standard net.

In addition, there are two classic versions of color for the polyhedron, when each of the adjacent faces is painted in its own color. The same colors do not border with each other.

We present two options for painting 20 faces of the icosahedron using five colors.

 

icosahedron five colors

icosahedron net

The first variant of the coloring of the icosahedron implies that each five vertex will have all five colors.

Download net:

icosahedron five colors

icosahedron net

The second variant of the coloring of the icosahedron provides the opposite faces of the same color.

Download net:

Icosahedron from the "Magic Edges" set

You can make an icosahedron model using assembling parts from the "Magic Edges" set.

 magic edges 12 regular polyhedra

Video. The build a polyhedron from the set:

 

Detailed assembly from Alexei Zhigulev (youtube channel - Origami)

Video. The rotation of the finished polyhedron:

Video. The rotation of all regular polyhedra

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