## Rg-Calculator of Polyhedra

Selected Polyhedra can consist of up to 3 different types of faces (equilateral n-polygonal or rhombic). Enter the type and number of the (different) polygonal faces, their common (equilateral) edge length (a) and either their common out-sphere(ro), mid-sphere(rm) or in-sphere(ri) radius originating in the centroid of the respective polyhedron, extending to the centroid of the faces (ri), of the edges (rm) or to the vertices (ro), respectively. In case of rhombic faces only the in-sphere-radius (ri) can be entered together with the ratio of the rhombic axes (0 < rx/ry < 1). The volume, surface, perimeter (sum of edge lengths), Radius of Gyration of the solid (Rgs), of the faces (Rgf) and of the edges (Rge) will be calculated. Units are arbitrary.

a: rx/ry:
r : in-radius out-radius mid-radius
n:
n:
n:

 Volume: Surface: Perimeter: Rg (solid): Rg (faces) Rg (edges):

## Radius of Gyration (Rg) of the 5 Platonic Solids

The values for the Volume (V), the Surface (S), the out-sphere (circum-sphere) radius (ro), the mid-sphere radius (rm), the in-sphere radius (ri), the Radius of Gyration of the solid (Rgs), of the faces (Rgf) and of the edges (Rge), the number and type of polygonal faces (n x p), number of edges (e) and vertices (v) of the 5 Platonic Solids with edge length a = 1 are compiled in the following table. Units are arbitrary.

 Solid Volume Surface ro rm ri Rg(s) Rg(f) Rg(e) faces e v Tetrahedron 0.11785 1.73205 0.61237 0.35355 0.20412 0.27386 0.35355 0.45644 4x3p 6 4 Hexahedron 1.00000 6.00000 0.86603 0.70711 0.50000 0.50000 0.64550 0.76376 6x4p 12 8 Octahedron 0.47140 3.46410 0.70711 0.50000 0.40825 0.38730 0.50000 0.57735 8x3p 12 6 Dodecahedron 7.66312 20.64573 1.40126 1.30902 1.11352 0.95448 1.23223 1.34047 12x5p 30 20 Icosahedron 2.18170 8.66025 0.95106 0.80902 0.75576 0.62666 0.80902 0.85898 20x3p 30 12

## Radius of Gyration (Rg) of the 13 Archimedean Solids

The values for the Volume (V), the Surface (S), the out-sphere (circum-sphere) radius (ro), the mid-sphere radius (rm), the Radius of Gyration of the solid (Rgs), of the faces (Rgf) and of the edges (Rge), the number and type of polygonal faces (n x p), number of edges (e) and vertices (v) of the 13 Archimedean Solids with edge length a = 1 are compiled in the following table. Units are arbitrary.

 Solid Volume Surface ro rm Rg(s) Rg(f) Rg(e) faces e v Truncated Tetrahedron 2.71058 12.12436 1.17260 1.06066 0.72005 0.91612 1.09924 4x3p 4x6p 18 12 Truncated Octahedron 11.31371 26.78461 1.58114 1.50000 1.08972 1.40452 1.52753 6x4p 8x6p 36 24 Truncated Cube 13.59966 32.43466 1.77882 1.70711 1.18122 1.51680 1.73134 8x3p 6x8p 36 24 Truncated Icosahedron 55.28773 72.60725 2.47802 2.42705 1.83259 2.36570 2.44416 12x5p 20x6p 90 60 Truncated Dodecahedron 85.03966 100.99076 2.96945 2.92705 2.12672 2.74285 2.94125 20x3p 12x10p 90 60 Cuboctahedron 2.35702 9.46410 1.00000 0.86603 0.64807 0.83497 0.91287 8x3p 6x4p 24 12 Icosidodecahedron 13.83553 29.30598 1.61803 1.53884 1.15787 1.49350 1.56568 20x3p 12x5p 60 30 Snub Cube 7.88948 19.85641 1.34371 1.24722 0.95857 1.23708 1.28019 32x3p 6x4p 60 24 Snub Dodecahedron 37.61665 55.28674 2.15584 2.09705 1.61125 2.07961 2.11683 80x3p 12x5p 150 60 Small Rhombicuboctah. 8.71405 21.46410 1.39897 1.30656 0.99132 1.27954 1.33808 8x3p 18x4p 48 24 Great Rhombicuboctah. 41.79899 61.75517 2.31761 2.26303 1.67650 2.16118 2.28137 12x4p 8x6p 6x8p 72 48 Small Rhombicosidodecah. 41.61532 59.30598 2.23295 2.17625 1.66620 2.15081 2.19531 20x3p 30x4p 12x5p 120 60 Great Rhombicosidodecah. 206.80340 174.29203 3.80239 3.76938 2.84666 3.67278 3.78041 30x4p 20x6p 12x10p 180 120

## Radius of Gyration (Rg) of 2 Rhombic Polyhedra

The values for the Volume (V), the Surface (S), the in-sphere radius (ri), the Radius of Gyration of the solid (Rgs), of the faces (Rgf) and of the edges (Rge), the ratio of the rhombic axes (rx/ry), the number of rhombic faces (f), edges (e) and vertices (v) of 2 Rhombic Polyhedra (Rhombic Dodecahedron and Rhombic Triacontahedron, which belong to the Catalan Solids and are Duals of 2 Archimedean Solids, the Cuboctahedron and the Icosidodecahedron, respectively) with edge length a = 1 are compiled in the following table. Units are arbitrary.

 Solid Volume Surface ri Rg(s) Rg(f) Rg(e) rx/ry f e v Rhombic Dodecahedron 3.07920 11.31371 0.81650 0.70711 0.91287 1.00000 0.70711 12 24 14 Rhombic Triacontahedron 12.31073 26.83282 1.37638 1.11205 1.43565 1.49257 0.61803 30 60 32

Author: M.Kriechbaum, TU-Graz (2013), e-mail: manfred.kriechbaum@tugraz.at