# Platonic Solids

Platonic solids, also known as regular solids or regular polyhedra, are 3-dimensional solids consisting of convex, regular polygons. As it is a regular polyhedron, each face is the same regular polygon, and the same number of polygons meets at each vertex.

They have been known since antiquity and were studied extensively by the Greeks. The Greek philosopher Plato associated Earth, air, water, and fire with solids. Earth was linked to the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron.

Further, significant studies on platonic solids by Euclid, Kepler, and others have led to our current understanding of them in more detail.

## Properties

The general properties found in platonic solids are:

1. Convex polyhedrons
2. All faces are polygonal, regular, and congruent
3. The same number of faces, which are polygons, meets at each vertex
4. Faces do not intersect except at their edges

## How Many Types of Platonic Solids are There

There are 5 types of platonic solids. They are: 1) tetrahedron, 2) Cube, 3) Octahedron, 4) Dodecahedron and 5) Icosahedron.

### Tetrahedron

It is a triangular pyramid. It has:

• 4 faces
• 6 edges
• 4 vertices
• 6 planes of symmetry
• All faces are equilateral triangles

### Cube

It is a regular hexahedron with identical square faces looking like a box. It has:

• 6 faces
• 12 edges
• 8 vertices
• All faces are square square-shaped
• The angles between any two faces are 90°
• The opposite faces are parallel
• Each face meets all other faces
• Each vertex meets the other 3 faces and 3 edges

### Octahedron

It is a polyhedron with faces shaped like an equilateral triangle. It has:

• 8 faces
• 12 edges
• 6 vertices
• 4 triangles meet at each vertex

### Dodecahedron

It is a polyhedron, with each face being a regular pentagon. It has:

• 12 faces
• 30 edges
• 20 vertices
• 3 pentagons meet at each vertex

### Icosahedron

It is a polyhedron with equilateral triangular faces. It has:

• 20 faces (highest for any platonic solids)
• 30 edges
• 12 vertices

## Existence of Platonic Solids

The existence of only 5 platonic solids can be proved using Euler’s formula. It is written as:

F + V – E = 2, here F = number of faces, V = number of vertices, and E = number of edges

Suppose we substitute the number of faces, edges, and vertices of any platonic solid in the above formula. In that case, it will make Euler’s formula accurate.

Let us prove it with the help of an example:

Consider the case of a cube. It has 6 faces, 12 edges, and 8 vertices.

Now, according to Euler’s formula

F + V – E = 2, here F = 6, V = 8, and E = 12

Substituting the values in the L.H.S. of the equation, we get

6 + 8 – 12

=> 2

Thus, a cube is a platonic solid (proved).

Prove how a tetrahedron is a platonic solid.

Solution:

As we know, according to Euler’s formula
F + V – E = 2, here F = 4, V = 4, and E = 6
Substituting the values in the L.H.S. of the equation, we get
4 + 4 – 6
=> 2
Thus, a tetrahedron is a platonic solid (proved).

## FAQs

1. Are all prisms platonic solids?

Ans. No

2. Which of the following platonic solids is also a pyramid?
a. tetrahedron b. Hexahedron c. octahedron d. icosahedron

Ans. a. tetrahedron

3. Are all rectangular prisms are platonic solids?

Ans. No

4. Which of the following platonic solids is also a cone?
a. tetrahedron b. Hexahedron c. octahedron d. d. none of the above

Ans. d. none of the above