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Dodecahedron

Dodecahedron

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

Therefore, the question, “what is a dodecahedron?” can be given the following definition: “A dodecahedron is a geometric solid of twelve faces, each of which is a regular pentagon.”

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

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

 

The regular dodecahedron is composed of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. Therefore, the sum of flat angles at each vertex is 324 °.
The dodecahedron has a center of symmetry - the center of the dodecahedron, 15 axes of symmetry and 15 planes of symmetry.

Mathematical characteristics of the dodecahedron

dodecahedron inscribed in a ball

A dodecahedron 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 dodecahedron

The radius of the described sphere of the dodecahedron

where a is the side length.

the ball is inscribed in the dodecahedron

The sphere can be inscribed inside the dodecahedron.

The radius of the inscribed sphere of the dodecahedron

 

The radius of the inscribed sphere of the dodecahedron

The surface area of the dodecahedron

The surface area of the dodecahedron

The surface area of the dodecahedron.

The surface area of a dodecahedron can be represented in the form of a net area.

The surface area can be defined as the area of one of the sides of the dodecahedron (this is the area of the regular pentagon) multiplied by 12. Or use the formula:

 

formula the surface area of the dodecahedron

The volume of the dodecahedron

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

formula The volume of the dodecahedron

Dodecahedron nets

You can make a dodecahedron 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, according to one of the versions of his treatise, did not ascribe the dodecahedron to any of the earth elements; but according to another version, he associated the dodecahedron with ether (emptiness). To build a model of this regular polyhedron, we chose yellow.

Dodekaehedron net

The figure shows the net of the dodecahedron:

Note that this is not the only option for a dodecahedron 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 draw your attention to two options for painting the dodecahedron using six and four colors.

 

The first option is six colors: download net Dodekaehedron six colors Dodecahedron net six colors
The second option is four colors: download the netDodecahedron four colors Dodecahedron net four colors

Video. Dodecahedron from the "Magic Edges" set

You can make a model of a dodecahedron using the parts from the "Magic Edges" set.

magic edges 12

Video. Build a polyhedron from the set:

Video. Rotation of the finished polyhedron:

Video. Rotation of all regular polyhedra

 

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