Regular polyhedra are also known as platonic solids, the five platonic solids can be viewed below; all of these are included in this page.

Tetrahedron

A tetrahedron is a polyhedron which is made of four triangular faces, three of which meet at each vertex. A regular tetrahedron has four triangles that are regular, or "equilateral", and it is one of the Platonic solids. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. With the tetrahedron the base is a triangle (any of the four faces can be the base), so a tetrahedron is also known as triangular pyramid. Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

Cube

A cube is a three-dimensional solid object which has six square faces, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism. It has cubical symmetry (also called octahedral symmetry). The cube has 3 classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.

A net of a Cube

(3 colours) | 2 2 2
D2h

(2 colours) 4 2 | 2
D4h

(1 colour) 3 | 4 2
Oh

The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into 6 identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is made.

Vertex: 8
Edges: 12
Faces: 6

For an amazing and educational experience visit http://polyhedra.org/poly/ This shows many different polygons which may be new and interesting to you.

Octahedron

Frame of an Octahedron

A net of an Octahedron

An octahedron is a regular polyhedron; it is a platonic solid which is made of eight equilateral triangles four of which meet at each vertex. The area A and the volume V of a regular octahedron of edge length a are:

Octahedron

A= 2√3a² ≈ 3.46410162a² V= ⅓√2a³ ≈ 0.471404521a³ Therefore the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

Vertex: 6
Edges: 12
Faces: 8

The octahedron can also be considered a rectified tetrahedron - and can be called a tetratetrahedron. This can be shown by a 2-color face model. This shows that the octahedron has tetrahedral symmetry.

## Tetrahedron

## A tetrahedron is a polyhedron which is made of four triangular faces, three of which meet at each vertex. A regular tetrahedron has four triangles that are regular, or "equilateral", and it is one of the Platonic solids. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. With the tetrahedron the base is a triangle (any of the four faces can be the base), so a tetrahedron is also known as triangular pyramid. Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

## Cube

## A cube is a three-dimensional solid object which has six square faces, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism. It has cubical symmetry (also called octahedral symmetry). The cube has 3 classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.A net of a Cube

(3 colours)

| 2 2 2D2h

(2 colours)

4 2 | 2D4h

(1 colour)

3 | 4 2Oh

## The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into 6 identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is made.

Vertex: 8

Edges: 12

Faces: 6

For an amazing and educational experience visit http://polyhedra.org/poly/ This shows many different polygons which may be new and interesting to you.

## Octahedron

An octahedron is a regular polyhedron; it is a platonic solid which is made of eight equilateral triangles four of which meet at each vertex.

The area

Aand the volumeVof a regular octahedron of edge lengthaare:A= 2√3a² ≈ 3.46410162a²

V= ⅓√2a³ ≈ 0.471404521a³

Therefore the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

Vertex: 6

Edges: 12

Faces: 8

## The octahedron can also be considered a rectified tetrahedron - and can be called a tetratetrahedron. This can be shown by a 2-color face model. This shows that the octahedron has tetrahedral symmetry.