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Icosahedron

Icosahedron

The ancient Greeks gave the polyhedron a name by the number of faces. “Ikosi” means twenty, “hedra” means a face (Icosahedron is a body 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 270 °.
The icosahedron has a center of symmetry - the center of the icosahedron, 15 axes of symmetry and 15 planes of symmetry.

Mathematical characteristics of icosahedron

Икосаэдр вписанный в сферу

An 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

 

Icosahedron R

where "a" - is the side length.

сфера вписанная в икосаэдр

The sphere can be inscribed inside the icosahedron.

The radius of the icosahedron inscribed sphere

 

Icosahedron r1

Icosahedron s

Icosahedron4 s

The surface area of the icosahedron can be represented as a net area.

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:

 

Icosahedron v

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

 

Icosahedron nets

Icosahedron can be made by yourself. Paper or cardboard is the most suitable option. For the assembly will require paper net - a single part with the lines of the folds.

plato 1 lg

Choose a color for a 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.

Ikosaehdr ris2ikos

The figure shows an icosahedron net:

Note that this is not the only net option.

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 - net.

In addition, there are two classic versions of the color of the polyhedron, when each of the adjacent faces is painted in its own color. Or use a certain number of colors coloring, and the same colors do not border with each other.

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

 

икосаэдр
развертка икосаэдра

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

Download net:

Ikosaehdr ris1ikos4
развертка икосаэдра

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

  Download net:

Video. Icosahedron from the set "Magic Edges"

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

12 obl 400

Video. Build a polyhedron from a set:

 

Video. Rotation of the finished polyhedron:

Video. Rotation of all regular polyhedra

 

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