As an example, the Radius of Gyration (Rg) of a Truncated Icosahedron (the shape of the C60-molecule) is being calculated. The polyhedral shape consists of 20 hexagons and 12 pentagons (60 vertices, 32 faces, 90 edges) with an unilateral edge-length

The parameters for the solid shape are entered as input into the Rg-calculator as shown below:

Parameter (Truncated Icosahedron) |
Value |

Volume | 55.288 |

Surface | 72.607 |

Perimeter | 90.000 |

out-sphere radius ro | 2.478 |

mid-sphere radius rm | 2.427 |

in-sphere radius ri (pentagons) | 2.327 |

in-sphere radius ri (hexagons) | 2.267 |

Rg of solid | 1.833 |

Rg of faces | 2.366 |

Rg of edges | 2.444 |

Rg of vertices ( = ro) | 2.478 |

As another example, the Radius of Gyration (Rg) of the Cuboctahedron (CO) and of two of its variations (Octahemioctahedron (OH) and Cubohemioctahedron (CH)) are being calculated. The polyhedral shape consists of 8 trigons and 6 tetragons (12 vertices, 14 faces, 24 edges) with an unilateral edge-length

The parameters for the solid shape are entered as input into the Rg-calculator as shown below:

Parameter (Cuboctahedron) |
Value |

Volume | 2.357 |

Surface | 9.464 |

Perimeter | 24.000 |

out-sphere radius ro | 1.000 |

mid-sphere radius rm | 0.866 |

in-sphere radius ri (trigons) | 0.816 |

in-sphere radius ri (tetragons) | 0.707 |

Rg (CO) | 0.648 |

Rg (OH) | 0.671 |

Rg (CH) | 0.632 |

Next, the Radius of Gyration (Rg) of the Small Rhombicuboctahedron (RCO) and of two of its variations (Small Cubicuboctahedron (CCO) and Small Rhombihexahedron (RH)) are being calculated. The polyhedral shape consists of 8 trigons and 18 tetragons (24 vertices, 26 faces, 48 edges) with an unilateral edge-length

The parameters for the solid shape are entered as input into the Rg-calculator as shown below:

Parameter (Small Rhombicuboctahedron) |
Value |

Volume | 8.714 |

Surface | 21.464 |

Perimeter | 48.000 |

out-sphere radius ro | 1.399 |

mid-sphere radius rm | 1.307 |

in-sphere radius ri (trigons) | 1.274 |

in-sphere radius ri (tetragons) | 1.207 |

Rg (RCO) | 0.991 |

Rg (CCO) | 0.926 |

Rg (RH) | 0.989 |

Last, the Radius of Gyration (Rg) of two variations of the Octahedron (Stella Octangula (SO) and Tetrahemihexahedron (THH)) are being calculated. The polyhedral shape (octahedron) consists of 8 trigons (6 vertices, 8 faces, 12 edges) with an unilateral edge-length

Parameter (Octahedron) |
Value |

Volume | 0.471 |

Surface | 3.464 |

Perimeter | 12.000 |

out-sphere radius ro | 0.707 |

mid-sphere radius rm | 0.500 |

in-sphere radius ri (trigons) | 0.408 |

Rg (O) | 0.387 |

Rg (SO) | 0.592 |

Rg (THH) | 0.387 |

A cube of edge length = 1 can be dissected into a central regular tetrahedron with edge length = √2 (rx=ry=rz) and 4 attached trigonal pyramids with edge length √2 (of the trigonal base, rx=ry) and height = 1/√3 (rz) attached to the 4 faces of the central tetrahedron. The distance between the center of mass of the tetrahedron to each center of mass of the 4 pyramids is √3/4 (rc). Calculation the Radius of Gyration of this cube (with edge length = 1) from these 5 subunits (with the proper geometry and coordinates) results in Rg=0.5 and V=1, (Rg(cube) = ∛V/2), using the Rg-calculator as shown below:

Alternatively a cube of edge length = 1 can also be dissected into 6 quadratic pyramids with edge length = 1 (of the quadratic base, rx=ry) and height = 1/2 (rz). The distance between the center of mass of the cube to each center of mass of the 6 pyramids is 3/8 (rc). Calculation the Radius of Gyration of this cube (with edge length = 1) from these 6 subunits (with the proper geometry and coordinates) results in Rg=0.5 and V=1, (Rg(cube) = ∛V/2), using the Rg-calculator as shown below: