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Hexahedron (Cube)

cube

The ancient Greeks gave the polyhedron a name by the number of faces. “Hexa” means six, “hedra” means a face (Hexahedron is a body with six faces).

Therefore, to the question “what is a hexahedron?”, We can give the following definition: “A hexahedron is a geometric solid of six faces, each of which is a regular quadrilateral (square)”.

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

 

The face of a polyhedron is a square The face of a polyhedron is a square. Each of the four fringe is 90 degrees.

Characteristics of the hexahedron (cube)

cube side

The number of sides at the face - 4

faces of the cubeTotal number of faces - 6
Face shape square
The number of edges adjacent to each vertex

The number of edges adjacent to each vertex - 3

Total number of cube vertices

Total number of vertices - 8

Total number of cube edges

Total number of edges - 12

parallel edgesEach edge (red) has three parallel edges (blue).
The number of pairs of parallel edges can be determined by multiplying the total number of edges by 3.
In a cube, 18 pairs of parallel edges.
perpendicular ribs
perpendicular ribs

Each edge (red) has 8 edges perpendicular to it (blue). To determine the number of pairs of perpendicular edges, you can multiply the total number of edges by 8 and divide by 2.
In total, the cube has 48 pairs of perpendicular edges.

 

crossed ribs

Each rib (red) has 4 ribs intersecting it.

Determine the number of pairs of crossed edges by multiplying the total number of edges by 4 and dividing by 2.

In total, the cube has 24 pairs of intersecting edges.

 

parallel faces

Number of pairs of parallel faces - 3

Distance between opposing edges

The distance between the opposite edges can be determined by the formula

formula Distance between opposing edges

,where "a" is the side length

The length of the cube diagonal

The length of the cube diagonal can be determined by the formula

formula The length of the cube diagonal

center of symmetry of a cubeThe cube has a center of symmetry.
_

The cube has 9 axes of symmetry.

  Three axes of symmetry are straight lines passing through the center of the parallel faces of the cube:

axis of rotation of the cube
axis of rotation of the cube
axis of rotation of the cube

The six axes of symmetry are the direct connecting centers of the opposite edges of the cube:

axis of rotation of the cube
axis of rotation of the cube
axis of rotation of the cube
axis of rotation of the cube
axis of rotation of the cube
axis of rotation of the cube

The cube has 9 planes of symmetry

  Three planes pass through the center parallel to the faces

 

cube plane of symmetry
cube plane of symmetry
cube plane of symmetry

Six planes pass through the center diagonally

cube plane of symmetry
cube plane of symmetry
cube plane of symmetry
cube plane of symmetry
cube plane of symmetry
cube plane of symmetry

cube placed in a sphere

A cube 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 cube

 

The radius of the described sphere of the cube

    where "a" -  is the side length.

The sphere can be inscribed inside the cube

The sphere can be inscribed inside the cube.

The radius of the cube's inscribed sphere

 

The radius of the cube's inscribed sphere

The sphere can be inscribed in a cube in such a way that it touches the surface of all the edges

The sphere can be inscribed in a cube in such a way that it touches the surface of all the edges of the cube. Such a sphere is called semi-inscribed in a cube.

The radius of the semi-written sphere can be determined by the formula:

The radius of the semi-written sphere

Surface area of the cube

Surface area of the cube

Surface area of the cube

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

The surface area can be defined as the area of one of the sides of the cube (this is the area of a regular quadrilateral - a square) multiplied by 6. Or use the formula:

 

formula Surface area of the cube

The volume of the cube

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

formula The volume of the cube

Cube nets

The cube can be made 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 hexahedron with the - ground, one of the basic "earthly" elements, so we chose a brown color to build a model of this regular polyhedron.

The figure shows a hexahedron net:

 

cube netTo 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 (pdf)
- if you use color cardboard for assembly - standard net (pdf)

Video. Cube from the "Magic Edges" set

You can make a model of hexahedron (cube) using the parts for assembly 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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