Mathematical properties of the Platonic solids

One can specify the following mathematical characteristics in each of the five Platonic solids:

1. The radius of the sphere circumscribing the polyhedron;

2. The radius of the sphere inscribed in the polyhedron;

3. The surface area of the polyhedron;

4. The volume of the polyhedron.

 

Tetrahedron:  All four faces are equilateral triangles

radius of a circumscribed sphere of tetrahedron

The radius of a circumscribed sphere of tetrahedron is

radius of a circumscribed sphere of tetrahedron

 ,where "a" is side length

radius of an inscribed sphere of tetrahedron

The radius of an inscribed sphere of tetrahedron is

radius of an inscribed sphere of tetrahedron

The surface area of tetrahedron

The surface area of tetrahedron

The surface area of tetrahedron

The surface area of tetrahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

volume of tetrahedron

The volume of tetrahedron is

volume of tetrahedron

Tetrahedron height

The height of the tetrahedron is determined by the following formula:

tetrahedron height formula

The distance to the center of the base of the tetrahedron is determined by the formula:

 distance to the center of the tetrahedron

 

Octahedron: All eight faces are equilateral triangles

radius of a circumscribed sphere of an octahedron

The radius of a circumscribed sphere
of an octahedron is

radius of a circumscribed sphere of an octahedron

,where "a" is side length

2 radius of an inscribed sphere of octahedron

The radius of an inscribed sphere of octahedron is

radius of an inscribed sphere of octahedron

The surface area of octahedron

The surface area of octahedron

The surface area of octahedron is

The surface area of octahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

volume of octahedron

The volume of octahedron is

volume of octahedron

 

Hexahedron (cube): All six faces are squares

radius of a circumscribed sphere of cube

The radius of a circumscribed sphere
of cube

radius of a circumscribed sphere of cube

,where "a" is side length

The radius of an inscribed sphere of cube

The radius of an inscribed sphere
of cube is

 

The radius of an inscribed sphere of cube

surface area of cube

surface area of cube

The surface area of cube is

surface area of cube

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of cube

The volume of cube is

The volume of cube

 

 

Dodecahedron: All 12 faces are regular pentagon


radius of a circumscribed sphere of dodecahedron

The radius of a circumscribed sphere
of dodecahedron

radius of a circumscribed sphere of dodecahedron

,where "a" is side length

radius of an inscribed sphere of dodecahedron

The radius of an inscribed sphere
of dodecahedron is

radius of an inscribed sphere of dodecahedron

surface area of dodecahedron

surface area of dodecahedron

The surface area of dodecahedron is

surface area of dodecahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

 

volume of dodecahedron

The volume of dodecahedron is

volume of dodecahedron

 

Icosahedron: All 20 faces are equilateral triangles

 

radius of a circumscribed sphere of icosahedron

The radius of a circumscribed sphere of icosahedron

radius of a circumscribed sphere of icosahedron

,where "a" is side length

radius of an inscribed sphere of icosahedron

The radius of an inscribed sphere of icosahedron is

radius of an inscribed sphere of icosahedron

surface area of icosahedron

surface area of icosahedron

The surface area of icosahedron is

surface area of icosahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

volume of icosahedron

the volume of icosahedron is

V volume of icosahedron