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


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, 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) faces f e v
Truncated Tetrahedron 0.51643 0.65705 0.78839 0.84100 4x3p 4x6p 8 18 12
Truncated Octahedron 0.48542 0.62564 0.68044 0.70432 6x4p 8x6p 14 36 24
Truncated Cube 0.49486 0.63546 0.72534 0.74523 8x3p 6x8p 14 36 24
Truncated Icosahedron 0.48105 0.62099 0.64158 0.65047 12x5p 20x6p 32 90 60
Truncated Dodecahedron 0.48362 0.62373 0.66884 0.67526 20x3p 12x10p 32 90 60
Cuboctahedron 0.48697 0.62740 0.68594 0.75141 8x3p 6x4p 14 24 12
Icosidodecahedron 0.48231 0.62212 0.65219 0.67399 20x3p 12x5p 32 60 30
Snub Cube 0.48151 0.62142 0.64307 0.67498 32x3p 6x4p 38 60 24
Snub Dodecahedron 0.48088 0.62066 0.63177 0.64341 80x3p 12x5p 92 150 60
Small Rhombicuboctahedron 0.48173 0.62180 0.65024 0.67983 8x3p 18x4p 26 48 24
Great Rhombicuboctahedron 0.48308 0.62273 0.65737 0.66781 12x4p 8x6p 6x8p 26 72 48
Small Rhombicosidodecahedron 0.48081 0.62066 0.63350 0.64436 20x3p 30x4p 12x5p 62 120 60
Great Rhombicosidodecahedron 0.48137 0.62107 0.63927 0.64299 30x4p 20x6p 12x10p 62 180 120
Sphere 0.48052 0.62035 0.62035 0.62035 000 000 000 000





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