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), of the edges (Rge) and of the vertices (Rgv) will be calculated for the entered parameters and also for the same shape where the volume is normalized to 1. 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): Rg (vertices):
Volume: Surface: Perimeter: Rg (solid): Rg (faces) Rg (edges): Rg (vertices):




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 and of their Duals (Catalan Solids).


The values for the Radius of Gyration of the solid (Rgs), of the faces (Rgf), of the edges (Rge) and of the vertices (Rgv) of the 13 Archimedean Solids and their corresponding Duals, the Catalan Solids, all with their respective volume normalized to V=1, are compiled in the following table. For comparison, also the values for a sphere with the same volume (=1) are given. Units are arbitrary.


Archimedean Solid Rg(s) Rg(f) Rg(e) Rg(v) Rg(v) Rg(e) Rg(f) Rg(s) Dual
Truncated Tetrahedron 0.51643 0.65705 0.78839 0.84100 0.84671 0.75184 0.65299 0.50581 Triakis Tetrahedron
Truncated Octahedron 0.48542 0.62564 0.68044 0.70432 0.71487 0.66002 0.62412 0.48344 Tetrakis Hexahedron
Truncated Cube 0.49486 0.63546 0.72534 0.74523 0.71843 0.68337 0.63082 0.48863 Triakis Octahedron
Truncated Icosahedron 0.48105 0.62099 0.64158 0.65047 0.65772 0.63364 0.62077 0.48085 Pentakis Dodecahedron
Truncated Dodecahedron 0.48362 0.62373 0.66884 0.67526 0.65267 0.64373 0.62229 0.48203 Triakis Icosahedron
Cuboctahedron 0.48697 0.62740 0.68594 0.75141 0.73482 0.68736 0.62748 0.48604 Rhombic Dodecahedron
Icosidodecahedron 0.48231 0.62212 0.65219 0.67399 0.66228 0.64641 0.62176 0.48161 Rhombic Triacontahedron
Snub Cube 0.48151 0.62142 0.64307 0.67498 0.66308 0.64913 0.62161 0.48150 Pentagonal Icositetrahedron
Snub Dodecahedron 0.48088 0.62066 0.63177 0.64341 0.63613 0.63199 0.62062 0.48073 Pentagonal Hexecontahedron
Small Rhombicuboctah. 0.48173 0.62180 0.65024 0.67983 0.67536 0.65001 0.62176 0.48161 Deltoidal Icositetrahedron
Great Rhombicuboctah. 0.48308 0.62273 0.65737 0.66781 0.66544 0.64201 0.62193 0.48174 Disdyakis Dodecahedron
Small Rhombicosidodecah. 0.48081 0.62066 0.63350 0.64436 0.64153 0.63206 0.62061 0.48072 Deltoidal Hexecontahedron
Great Rhombicosidodecah. 0.48137 0.62107 0.63927 0.64299 0.63669 0.62992 0.62077 0.48085 Disdyakis Triacontahedron
Dual Rg(s) Rg(f) Rg(e) Rg(v) Rg(v) Rg(e) Rg(f) Rg(s) Catalan Solid
Sphere 0.48052 0.62035 0.62035 0.62035 0.62035 0.62035 0.62035 0.48052 Sphere





Author: M.Kriechbaum, TU-Graz (2014), e-mail: manfred.kriechbaum@tugraz.at