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

The radius of a circumscribed sphere of the tetrahedron is:

, where "a" is the side length

The radius of an inscribed sphere of the tetrahedron is:



For clarity, the surface area of a tetrahedron can be represented as a shape net area.
The tetrahedron surface area can be defined as the area of one of the tetrahedron's sides. This is the area of a regular triangle, multiplied by 4. Or use the formula:


The volume of a tetrahedron is:


The following formula determines the height of the tetrahedron:

The formula determines the distance to the center of the base of the tetrahedron:

Octahedron: All eight faces are equilateral triangles

The radius of a circumscribed sphere
of an octahedron is:

, where "a" is the side length

The radius of an inscribed sphere of the octahedron is:



For clarity, the surface area of an octahedron can be represented as a shape net area.
The octahedron surface area can be defined as the area of one of the octahedron's sides. This is the area of a regular triangle, multiplied by 8. Or use the formula:


The volume of the octahedron is:

Hexahedron (cube): All six faces are squares

The radius of a circumscribed sphere
of cube:

, where "a" is the side length

The radius of an inscribed sphere
of the cube is:



For clarity, the surface area of a cube can be represented as a shape net area.
The cube surface area can be defined as the area of one of the cube's sides. This is the area of a regular quadrilateral (square), multiplied by 6. Or use the formula:


The volume of the cube is:

Dodecahedron: All 12 faces are regular pentagons

The radius of a circumscribed sphere
of dodecahedron:

, where "a" is the side length

The radius of an inscribed sphere
of dodecahedron is



For clarity, the surface area of a dodecahedron can be represented as a shape net area.
The dodecahedron surface area can be defined as the area of one of the dodecahedron's sides. This is the area of a regular pentagon, multiplied by 12. Or use the formula:


The volume of the dodecahedron is:

Icosahedron: All 20 faces are equilateral triangles

The radius of a circumscribed sphere of the icosahedron:

, where "a" is the side length

The radius of an inscribed sphere of the icosahedron is:



For clarity, the surface area of an icosahedron can be represented as a shape net area.
The icosahedron surface area can be defined as the area of one of the icosahedron's sides. This is the area of a regular triangle, multiplied by 20. Or use the formula:


the volume of the icosahedron is:
