**Platonic solids**

This animation demonstrates the five regular three-dimensional (or Platonic) solids, the best known of which is the cube.

**Mathematics**

**Keywords**

Platonic solid, tetrahedron, cube, octahedron, dodecahedron, icosahedron, dual, Pythagoras, Aristotle, geometry, solid geometry, mathematics

**Related items**

### Scenes

In Euclidean geometry, a Platonic solid is a **regular**, convex **polyhedron** composed of congruent faces that are regular polygons, with the same number of faces meeting at each vertex.

There are **five** **Platonic solids**, each is named according to its number of faces.

Tetrahedron: **4** faces

Hexahedron: **6** faces

Octahedron: **8** faces

Dodecahedron: **10** faces

Icosahedron: **20** faces

Polyhedra are associated into pairs called **duals**. In the dual of a polyhedron its vertices correspond to the faces of the other: the centers of each face are connected to the centers of the neighboring faces. The duals of regular polyhedrons are also regular polyhedrons, therefore Platonic solids are arranged into **pairs**.

You can view the dual of the solids by clicking the button ´Dual´.

A **dodecahedron** is a regular polyhedron composed of congruent, regular **pentagonal** faces.

Faces: **12**

Edges: **30**

Vertices: **20**

Dual: **icosahedron**

Angle between two faces: **~116°33’55.84”**

Number of edges meeting at each vertex: **3**

Space diagonals: **100**

Prepare a dodecahedron with unit edge lengths (Animation).

Take a cube in which the length of the edges is the golden ratio (t). Take also 3 congruent rectangles in which the shorter edges are 1 unit long, the longer edges **1+t** long. Place the rectangles inside the cube in a way that their centers fall on the center of the cube and any two of them intersect at right angles, with the plane of all three being parallel to one pair of faces of the cube. Now connect the vertices of the rectangles with the two nearest vertices of the cube. These lines, together with the shorter edges of the rectangles form the frame of a dodecahedron with unit edge lengths.

An **icosahedron** is a regular polyhedron composed of congruent, regular **triangular** faces.

Faces: **20**

Edges: **30**

Vertices: **12**

Dual: **dodecahedron**

Angle between two faces: **~138°11’22.87”**

Number of edges meeting at each vertex: **5**

Space diagonals: **36**

Prepare an icosahedron with unit edge lengths. (Animation)

Take 3 congruent golden rectangles. These are rectangles in which the ratio of the length of the edges is the golden ratio: ** a+b : a = a : b**. Let the shorter edge of these golden rectangles be of unit length, then the longer edges will be t long (where t is the golden ratio). Place the rectangles in a way that their centers coincide and any two of them intersect at right angles. Now connect the vertices of each rectangle with the two nearest vertices of the 2 other rectangles. These lines, together with the shorter edges of the rectangles form the frame of an icosahedron with unit edge lengths.

### Related items

#### Cube

This animation demonstrates the components (vertices, edges, diagonals and faces) of the cube, one of the Platonic solids.

#### Cylindrical solids

This animation demonstrates various types of cylindrical solids as well as their lateral surfaces.

#### Euler's polyhedron formula

The theorem formulated by Leonhard Euler describes one of the basic properties of convex polyhedra.

#### Perimeter, area, surface area and volume

This animation presents the formulas to calculate the perimeter and area of shapes as well as the surface area and volume of solids.

#### Regular square pyramid

A regular square pyramid is a right pyramid with a square base and four triangular faces.

#### Sphere

A sphere is the set of points which are all within the same distance from a given point in space.

#### Volume of a tetrahedron

To calculate the volume of a tetrahedron we start by calculating the volume of a prism.