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 n (>1) of the (different) n-polygonal (n=3/4/5/6/8/10) faces, their common (equilateral) edge length (a) and either their common out-sphere(ro) or mid-sphere(rm) or in-sphere(ri) radius originating in the centroid of the respective polyhedron (not of the polygon!), extending to the centroid of the faces ri (ri > 0) or of the edges rm (rm > a/(2*tan(Pi/n)) ) or of the vertices ro (ro > a/(2*sin(Pi/n)) ), respectively. Note, that the Platonic Solids (consisting of 1 type of regular polygons) have one common ro, rm, ri for the polygon (all three values are different though), the Archimedean Solids (consisting of 2 or 3 different types of regular polygons) have one common ro and rm but different ri for the different polygons (ri for the different polygons will be also calculated and displayed) The Catalan Solids (consisting of 1 type of irregular polygons with different edge lengths) have one common ri but different ro for the polygon. The volume, convex surface (sum of polygon areas), perimeter (sum of polygon 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. Examples are shown here.


a:
r : in-radius out-radius mid-radius
n: ri:
n: ri:
n: ri:




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), of the edges (Rge) and of the vertices (Rgv, not listed here as Rgv = ro), 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


The values for the same parameters as above for the Platonic Solids normalized to the same volume V=1 (with the respective edge lengths a) compared to a sphere with V=1 and r=0.62035. Units are arbitrary.


Solid Edge (a) Surface ro rm ri Rg(s) Rg(f) Rg(e)
Tetrahedron 2.03965 7.20562 1.24902 0.72112 0.41634 0.55858 0.72112 0.93097
Hexahedron 1.00000 6.00000 0.86603 0.70711 0.50000 0.50000 0.64550 0.76376
Octahedron 1.28490 5.71911 0.90856 0.64245 0.52456 0.49764 0.64245 0.74184
Dodecahedron 0.50722 5.31161 0.71075 0.66396 0.56480 0.48413 0.62501 0.67992
Icosahedron 0.77103 5.14835 0.73329 0.62377 0.58271 0.48317 0.62377 0.66229
Sphere - 4.83599 0.62035 0.62035 0.62035 0.48052 0.62035 0.62035




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, together with the number of the faces (f), edges (e) and vertices (v) for each polyhedron. For comparison, also the Rg-values for a sphere with the same volume (V=1) are given. Units are arbitrary. Source of shape images: Wikipedia.


Shape Archimedean Solid Rg(s) Rg(f) Rg(e) Rg(v) f e v Rg(v) Rg(e) Rg(f) Rg(s) Dual
Truncated Tetrahedron 0.51643 0.65705 0.78839 0.84100 8 18 12 0.84671 0.75184 0.65299 0.50581 Triakis Tetrahedron
Truncated Octahedron 0.48542 0.62564 0.68044 0.70432 14 36 24 0.71487 0.66002 0.62412 0.48344 Tetrakis Hexahedron
Truncated Cube 0.49486 0.63546 0.72534 0.74523 14 36 24 0.71843 0.68337 0.63082 0.48863 Triakis Octahedron
Truncated Icosahedron 0.48105 0.62099 0.64158 0.65047 32 90 60 0.65772 0.63364 0.62077 0.48085 Pentakis Dodecahedron
Truncated Dodecahedron 0.48362 0.62373 0.66884 0.67526 32 90 60 0.65267 0.64373 0.62229 0.48203 Triakis Icosahedron
Cuboctahedron 0.48697 0.62740 0.68594 0.75141 14 24 12 0.73482 0.68736 0.62748 0.48604 Rhombic Dodecahedron
Icosidodecahedron 0.48231 0.62212 0.65219 0.67399 32 60 30 0.66228 0.64641 0.62176 0.48161 Rhombic Triacontahedron
Snub Cube 0.48151 0.62142 0.64307 0.67498 38 60 24 0.66308 0.64913 0.62161 0.48150 Pentagonal Icositetrahedron
Snub Dodecahedron 0.48088 0.62066 0.63177 0.64341 92 150 60 0.63613 0.63208 0.62062 0.48073 Pentagonal Hexecontahedron
Small Rhombicuboctah. 0.48173 0.62180 0.65024 0.67983 26 48 24 0.67536 0.65001 0.62176 0.48161 Deltoidal Icositetrahedron
Great Rhombicuboctah. 0.48308 0.62273 0.65737 0.66781 26 72 48 0.66544 0.64201 0.62193 0.48174 Disdyakis Dodecahedron
Small Rhombicosidodecah. 0.48081 0.62066 0.63350 0.64436 62 120 60 0.64153 0.63206 0.62061 0.48072 Deltoidal Hexecontahedron
Great Rhombicosidodecah. 0.48137 0.62107 0.63927 0.64299 62 180 120 0.63669 0.62992 0.62077 0.48085 Disdyakis Triacontahedron
Shape Dual Rg(s) Rg(f) Rg(e) Rg(v) v e f Rg(v) Rg(e) Rg(f) Rg(s) Catalan Solid
Sphere 0.48052 0.62035 0.62035 0.62035 000 000 000 0.62035 0.62035 0.62035 0.48052 Sphere





Author: M.Kriechbaum, TU-Graz (2014, 2023), e-mail: manfred.kriechbaum@tugraz.at