Polyhedra: Math Tables for Platonic And Archimedean Solids

Polyhedra Tables of Platonic And Archimedean Solids

Names, Symmetries, Numbers of Polygons, Faces, Edges, Vertices, Surface Areas, Volumes, Dihedral Angles, Central Angles, Sphere Ratios of Insphere, Intersphere, Circumsphere Radius and Edges, Face Angles for Corresponding Face Components

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type key: p=platonic, a=archimedean faces dihedral angles between polygons with this many sides
type: # name(s) symmetry # of triangles : 3 # of squares : 4 # of pentagons : 5 # of hexagons : 6 # of octagons : 8 # of decagons : 10 total # of faces total # edges total # vertices (a.k.a. apices) total surface area (edge length = 1) total surface area (circumradius = 1) total surface area (inradius = 1) volume * 3-3 4-4 5-5 3-4 3-5 3-6> 3-8 3-10 4-5 4-6 4-8 4-10 5-6 6-6 6-8 6-10 8-8 10-10 central angle r/e =
circumradius/
edge length
e/r =
edge length/
circumradius
r/e =
inradius/
edge length
e/r =
edge length/
inradius
rho/e =
intersphere/
edge length
e/rho =
edge length/
intersphere
r/r =
circumsphere/
insphere
r/r =
insphere/
circumsphere
r/rho =
circumsphere/
intersphere
rho/r =
intersphere/
circumsphere
r/rho =
insphere/
intersphere
rho/r =
intersphere/
insphere
p 1 tetrahedron tetrahedral 4 4 6 4 1.732050808 2.828427125 8.485281374 0.117851130 70° 32′ 109° 28′ 0.612372435696 1.632993161855 0.204124145232 4.898979485566 0.353553390593 2.828427124746 3.000000000000 0.333333333333 1.732050807569 0.577350269190 0.577350269190 1.732050807569
p 2 hexahedron (cube) 2,3,4-fold 6 6 12 8 6.000000000 6.928203230 12.000000000 1.000000000 90° 70° 32′ 0.866025403784 1.154700538379 0.500000000000 2.000000000000 0.707106781187 1.414213562373 1.732050807569 0.577350269190 1.224744871392 0.816496580928 0.707106781187 1.414213562373
p 3 octahedron 2,3,4-fold 8 8 12 6 3.464101615 4.898979485 8.485281374 0.471404521 109° 28′ 90° 0.707106781187 1.414213562373 0.408248290464 2.449489742783 0.500000000000 2.000000000000 1.732050807569 0.577350269190 1.414213562373 0.707106781187 0.816496580928 1.224744871392
p 4 dodecahedron 2,3,5-fold 12 12 30 20 20.645728806 14.733704195 18.541019661 7.663118961 116° 34′ 41° 49′ 1.401258538444 0.713644179546 1.113516364412 0.898055953159 1.309016994375 0.763932022500 1.258408572365 0.794654472292 1.070466269319 0.934172358963 0.850650808352 1.175570504585
p 5 icosahedron 2,3,5-fold 20 20 30 12 8.660254038 9.105929973 11.458980337 2.181694991 138° 11′ 63° 26′ 0.951056516295 1.051462224238 0.755761314076 1.323169076499 0.809016994375 1.236067977500 1.258408572365 0.794654472292 1.175570504585 0.850650808352 0.934172358963 1.070466269319
a 1 truncated octahedron (mecon) 2,3,4-fold 6 8 14 36 24 26.784609689 16.940074571 18.822305079 11.31370850 125° 16′ 109° 28′ 36° 52′ 1.581138830084 0.632455532034 1.423024947076 0.702728368926 1.500000000000 0.666666666667 1.111111111111 0.900000000000 1.054092553389 0.948683298051 0.948683298051 1.054092553389
a 2 cuboctahedron (dymaxion) 2,3,4-fold 8 6 14 24 12 9.464101615 9.464101615 12.618802153 2.357022604 125° 16′ 60° 1.000000000000 1.000000000000 0.750000000000 1.333333333333 0.866025403784 1.154700538379 1.333333333333 0.750000000000 1.154700538379 0.866025403784 0.866025403784 1.154700538379
a 3 truncated cuboctahedron 2,3,4-fold 12 8 6 26 72 48 61.755172435 26.646048347 27.946789492 41.79898987 144° 44′ 135° 125° 16′ 24° 55′ 2.317610912893 0.431478810545 2.209741210257 0.452541680156 2.263033438454 0.441884765381 1.048815536469 0.953456509013 1.024116954488 0.976450976247 0.976450976247 1.024116954488
a 4 snub cube 2,3,4-fold 32 6 38 60 24 19.856406460 14.777263402 17.152165352 7.889477400 153° 14′ 142° 59′ 43° 41′ 1.343713373745 0.744206331156 1.157661790956 0.863810145426 1.247223167994 0.801781129201 1.160713244786 0.861539234167 1.077364026124 0.928191377986 0.928191377986 1.077364026124
a 5 (small) rhombicuboctahedron 2,3,4-fold 8 18 26 48 24 21.464101615 15.342829357 17.589734695 8.714045208 144° 44′ 135° 41° 53′ 1.398966325966 0.714813488673 1.220262953798 0.819495500448 1.306562964876 0.765366864730 1.146446609407 0.872260419103 1.070722470768 0.933948831094 0.933948831094 1.070722470768
a 6 truncated cube 2,3,4-fold 8 6 14 36 24 32.434664361 18.233771763 19.797982086 13.59966329 125° 16′ 90° 32° 39′ 1.778823645664 0.562169275430 1.638281326807 0.610395774912 1.707106781187 0.585786437627 1.085786437627 0.920991426441 1.042010766560 0.959682982261 0.959682982261 1.042010766560
a 7 truncated icosahedron (soccer ball) 2,3,5-fold 12 20 32 90 60 72.607253029 29.300527163 30.544061106 55.28773076 142° 37′ 138° 11′ 23° 17′ 2.478018659068 0.403548212335 2.377131605984 0.420675068003 2.427050983125 0.412022659167 1.042440667917 0.959287210080 1.020999837373 0.979432085486 0.979432085486 1.020999837373
a 8 icosidodecahedron 2,3,5-fold 20 12 32 60 30 29.305982843 18.112093471 20.024238056 13.83552594 142° 37′ 36° 1.618033988750 0.618033988750 1.463525491562 0.683281573000 1.538841768588 0.649839392466 1.105572809000 0.904508497187 1.051462224238 0.951056516295 0.951056516295 1.051462224238
a 9 truncated icosidodecahedron 2,3,5-fold 30 20 12 62 180 120 174.292030327 45.837440154 46.643971373 206.8033989 159° 6′ 148° 17′ 142° 37′ 15° 6′ 3.802394499851 0.262992175073 3.736646456083 0.267619645517 3.769377127922 0.265295821050 1.017595468167 0.982708778963 1.008759370795 0.991316689541 0.991316689541 1.008759370795
a 10 snub dodecahedron 2,3,5-fold 80 12 92 150 60 55.286744956 25.645137056 27.103030805 37.61664996 164° 11′ 152° 56′ 26° 49′ 2.155837375116 0.463856880645 2.039873154954 0.490226560201 2.097053835252 0.476859479327 1.056848740756 0.946209198569 1.028031488212 0.972732850566 0.972732850566 1.028031488212
a 11 (small) rhombicosidodecahedron 2,3,5-fold 20 30 12 62 120 60 59.305982843 26.559470348 27.961449293 41.61532378 159° 6′ 148° 17′ 25° 52′ 2.232950509416 0.447837959589 2.120991019518 0.471477715274 2.176250899483 0.459505841095 1.052786404500 0.949860290488 1.026053801952 0.974607762378 0.974607762378 1.026053801952
a 12 truncated dodecahedron 2,3,5-fold 20 12 32 90 60 100.990760142 34.009932348 35.002328800 85.03966456 142° 37′ 116° 34′ 116° 34′ 19° 24′ 2.969449015863 0.336762811773 2.885258312920 0.346589418189 2.927050983125 0.341640786500 1.029179606750 0.971647702152 1.014484897251 0.985721919281 0.985721919281 1.014484897251
a 13 truncated tetrahedron tetrahedral 4 4 8 18 12 12.124355652 10.339685242 12.637393073 2.710575995 109° 28′ 70° 32′ 50° 28′ 1.172603939956 0.852802865422 0.959403223600 1.042314613294 1.060660171780 0.942809041582 1.222222222222 0.818181818182 1.105541596785 0.904534033733 0.904534033733 1.105541596785
face angles for corresponding face components 60° 90° 108° 120° 135° 144° * in multiples of edge length cubed