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Octahedron

Octahedron

The ancient Greeks gave the polyhedron a name according to the number of faces. "Octo" means eight, "hedra" means the face (octahedron - a solid with eight faces).

Therefore, the question - "What is an octahedron?", We can give the following definition: "An octahedron is a geometric solid of eight faces, each of which is a right triangle."

The polyhedron belongs to regular polyhedra and is one of the five Platonic solids.

The octahedron 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 8;
    
The number of edges adjacent to the top is 4;
    
The total number of vertices is 6;
    
The total number of edges is 12;

 

The regular octahedron is composed of eight equilateral triangles. Each vertex of the octahedron is a vertex of four triangles. Therefore, the sum of flat angles at each vertex is 240 °.

The octahedron has a center of symmetry, the center of the octahedron has 9 axes of symmetry and 9 planes of symmetry.

The octahedron can be represented as two regular pyramids with a quadrilateral base connected to each other through this base.

 

Mathematical characteristics of the octahedron

The radius octahedron of the described sphere

The octahedron 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 octahedron is determined by the formula:

 

formula The radius of the described sphere of the octahedron

, where "a" is the side length.

sphere inscribed inside the octahedron

The sphere can be inscribed inside the octahedron.

The radius of the octahedron inscribed sphere is determined by the formula:

 

formula sphere inscribed inside the octahedron

The surface area of the octahedron

The surface area of the octahedron

The surface area of the octahedron

The surface area of an octahedron 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 octahedron (this is the area of a regular triangle) multiplied by 8. Or use the formula: formula The surface area of the octahedron

The volume of the octahedron

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

formula The volume of the octahedron

 

Octahedron nets

You can make the octahedron 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 octahedron with the "earth" element - air; therefore, to build a model of this regular polyhedron, we chose gray.

octahedron net

The figure shows the octahedron net:

Note that this is not the only option for a octahedron 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 use color cardboard for assembly - standard net.

The classic version of the coloring involves the color of the octahedron in four different colors, and in such a way that each face has its own color that is different from the next and only the opposite faces that are not in contact with each other are painted in the same color.

The coloring option is presented in the figure. You can download the net with the appropriate coloring of the faces.

octahedron four colorsoctahedron net four colors

Video. Octahedron from the "Magic Edges" set

You can make a model of the octahedron using the parts to assemble from the "Magic Edges" set.

magic edges 12 platonic solids

Video. Build a polyhedron from a set:

Video. Rotation of the finished polyhedron:

 

Video. Rotation of regular polyhedra

 

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