Polyhedra Tables of Platonic And Archimedean Solids

Images of the Five Platonic Solids

Images of the Five Platonic Solids

Here are fold-up patterns for the Platonic Solids.


Images of the Thirteen Archimedean Solids

Images of the Thirteen Archimedean Solids

Here are fold-up patterns for the Archimedean Solids.


Update to page 243 of Sacred Geometry Design Sourcebook

Update to page 243 of Sacred Geometry Design Sourcebook

Note This update only applies if you a copy of Sacred Geometry Design Sourcebook that was printed before about 1999; check page 243 and look for the volume of the Snub Dodecahedron; if you don't see: "37.61665", then copy the graphic below and paste it over the corresponding section on page 243 to provide more accuracy.

A number of years ago, realizing that I needed a few more digits to complete the desired 6-decimal place accuracy for a few of the more complicated archimedeans (particularly the snub dodecahedron), I held a book give-away contest to the first person who could provide the missing digits of accuracy, and show how these were derived. (When I find his email in my archives, I'll post it here!) Johan of Delfgauw, Netherlands, was the contest winner (a true math enthusiast and a a student of Industrial Design Engineering at the Technical University of Delft. Johan won a free copy of my Sacred Geometry Design Sourcebook.



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


Polyhedra Elements
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'
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