## The icosahedron.

The icosahedron is one of five types of regular polyhedra, has 20 faces (triangular), 30 edges, 12 vertices (at every vertex converge 5 ribs).

The icosahedron, as the correct polyhedron

The basic formula of the icosahedron

## The icosahedron, as the correct polyhedron:

**The icosahedron** (from al-Greek. εἴκοσι “twenty”; ἕδρον “seat”, “base”) is one of the five types of regular polyhedra, has 20 faces (triangular), 30 edges, 12 vertices (at every vertex converge 5 ribs).

If a is **the edge length of the icosahedron**, then its volume is V = 5/12 · a3 · (3+51/2) ≈ 2,1817 · a3.

**The icosahedron** has 59 stellate forms.

The vertices of the inscribed icosahedron lie in four parallel planes.

## The basic formula of the icosahedron:

The area of the surface S, the volume V of a regular icosahedron with edge length a, and the radii of the inscribed and circumscribed spheres are calculated by the formulas:

**The surface area of the icosahedron**: S = 5 · a2 · 31/2.

**The volume of the icosahedron**: V = 5/12 · a3 · (3+51/2).

**The radius of the inscribed sphere of the icosahedron**: R = 1/12 · (42+18 · (5·and)1/2)1/2 = 1/4 · 31/2 · (3+51/2) · and.

**The radius describes the sphere of the icosahedron**: R = 1/4 · (2 · (5+51/2))1/2 · a.

## Properties of the icosahedron:

– **the dihedral angle between any two adjacent faces of an icosahedron** is equal to arccos(-√5/3) = 138,189685°;

– corporal angle at the vertex of an icosahedron is equal to 2·π – 5·arcsin(2/3) ≈ 2,63455 cp (steradian);

**all twelve vertices of the icosahedron lie in four parallel planes, forming in each right triangle;**

ten vertices of the icosahedron lie in two parallel planes, forming them in two regular Pentagon and the remaining two are opposite to each other and lie on the two ends of the described diameter of the sphereperpendicular to these planes;

– **the icosahedron can be inscribed** **in a cube**, with six mutually perpendicular edges of the icosahedron are located respectively on the six faces of the cube, the remaining 24 edges inside the cube, all twelve vertices of the icosahedron will lie on the six faces of the cube;

in an icosahedron can be inscribed tetrahedron so that four vertices of the tetrahedron will be combined with the four vertices of an icosahedron;

**the icosahedron can be inscribed in a dodecahedron**, with vertices of the icosahedron are aligned with the centers of the faces of the dodecahedron;

in an icosahedron can be inscribed dodecahedron with a combination of the vertices of the dodecahedron and the centers of faces of the icosahedron;

**– the truncated icosahedron can be obtained by cutting off the 12 vertices with the formation of facets in the form of regular pentagons. ** The number of vertices of the new polyhedron is increased 5 times (12×5=60), 20 triangular faces become regular hexagons (all faces become 20+12=32), and the number of ribs increases to 30+12×5=90;

– to collect a model of an icosahedron can be using 20 equilateral triangles.

**– it is impossible to collect correct icosahedron from tetrahedra**, as described radius of the sphere around the icosahedron, respectively, and the length of a side edge (from the top to the center of such Assembly) of a tetrahedron is less than the edges of the icosahedron.

Source: https://ru.wikipedia.org/wiki/Правильный_икосаэдр

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