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# 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 tetrahedron is ,where "a" is side length The radius of an inscribed sphere of tetrahedron is   The surface area of the tetrahedron can be represented in the form of a surface area shape net. The surface area can be defined as the area of one of the sides of the tetrahedron - this is the area of a regular triangle, multiplied by 4. Or use the formula:  The volume of tetrahedron is  The height of the tetrahedron is determined by the following formula: The distance to the center of the base of the tetrahedron is determined by the formula: ## Octahedron: All eight faces are equilateral triangles The radius of a circumscribed sphere
of an octahedron is ,where "a" is side length The radius of an inscribed sphere of octahedron is   The surface area of an octahedron can be represented in the form of a surface area shape net. The surface area can be defined as the area of one of the sides of the octahedron - this is the area of a regular triangle, multiplied by 8. Or use the formula:  The volume of octahedron is ## Hexahedron (cube): All six faces are squares The radius of a circumscribed sphere
of cube ,where "a" is side length The radius of an inscribed sphere
of cube is   The surface area of the cube can be represented in the form of the surface area shape net. The cube surface area can be defined as the area of one of the sides of the cube - this is the area of a regular quadrilateral (square), multiplied by 6. Or use the formula:  The volume of cube is ## Dodecahedron: All 12 faces are regular pentagon The radius of a circumscribed sphere
of dodecahedron ,where "a" is side length The radius of an inscribed sphere
of dodecahedron is   The surface area of a dodecahedron can be represented in the form of surface area shape net. The surface area can be defined as the area of one of the sides of the dodecahedron (this is the area of the regular pentagon) multiplied by 12. Or use the formula:  The volume of dodecahedron is ## Icosahedron: All 20 faces are equilateral triangles The radius of a circumscribed sphere of icosahedron ,where "a" is side length The radius of an inscribed sphere of icosahedron is   The surface area of the icosahedron can be represented as a surface area shape net. The surface area can be defined as the area of one of the sides of the icosahedron (this is the area of a regular triangle) multiplied by 20. Or use the formula:  the volume of icosahedron is ### Popular

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