Properties of polyhedra
Introduction
This is a list of distances, areas, volumes and angles regarding some regular convex polyhedra with all edges having a length of one. Notes:
- faces: T=triangle, S=square, P=pentagon, H=hexagon, O=octagon, D=decagon
- circumradius R = radius of circumsphere (touches each vertex)
- midradius ρ = radius of midsphere (touches centre of each edge)
- inradius rn = radius of insphere of n-gonal faces (touches centre of each n-gon)
- this page is under construction
Platonic solids
Five solids. Three types of faces: triangle, square, pentagon. One type of face per solid.
| Name | Faces | Vertices | Circumradius | Midradius | Inradius | Surface area | Volume | Dihedral angle |
|---|---|---|---|---|---|---|---|---|
| tetrahedron, triangular pyramid, disphenoid, digonal antiprism |
4T | 4 | √6 / 4 | √2 / 4 | √6 / 12 | √3 | √2 / 12 | arccos(1 / 3) = arctan(2√2) |
| cube, square prism |
6S | 8 | √3 / 2 | √2 / 2 | 1 / 2 | 6 | 1 | 90° |
| octahedron, square bipyramid, triangular antiprism |
8T | 6 | √2 / 2 | 1 / 2 | √6 / 6 | 2√3 | √2 / 3 | arccos(−1 / 3) |
| dodecahedron | 12P | 20 | √3 × (1 + √5) / 4 | (3 + √5) / 4 | √(250 + 110√5) / 20 | 3√(25 + 10√5) | (15 + 7√5) / 4 | arccos(−√(5) / 5) |
| icosahedron | 20T | 12 | √(10 + 2√5) / 4 | (1 + √5) / 4 | √3 × (3 + √5) / 12 | 5√3 | 5(3 + √5) / 12 | arccos(−√(5) / 3) |
Sources:
- MathWorld: regular tetrahedron, cube, regular octahedron, regular dodecahedron, regular icosahedron
- WolframAlpha
Archimedean solids
13 solids. Two or three types of faces per solid. No solid has both pentagons and octagons, both pentagons and decagons, or both octagons and decagons. Snub cube and snub dodecahedron are chiral (different from their mirror images).
Constants used in the table:
- the golden ratio: φ = (1 + √5) / 2
- the tribonacci constant (used in snub cube): t = (1 + ∛(19 + 3√33) + ∛(19 − 3√33)) / 3 ≈ 1.83928676
- used in snub dodecahedron: u = real zero of x3 + 2x2 − φ2 = (∛(44 + 12φ(9 + √(81φ − 15))) + ∛(44 + 12φ(9 − √(81φ − 15))) − 4) / 6 ≈ 0.943151259
- used in snub dodecahedron: v = ∛(4φ + 4√(φ − 5 / 27)) / 2 + ∛(4φ − 4√(φ − 5 / 27)) / 2 ≈ 1.71556150
- used in snub dodecahedron: w = ∛(54(1 + √5) + 6√(102 + 162√5)) + ∛(54(1 + √5) − 6√(102 + 162√5)) ≈ 10.2933690
| Names | Faces | Vertices | Circumradius | Midradius | Inradius | Surface area | Volume |
|---|---|---|---|---|---|---|---|
| truncated tetrahedron | 4T+4H | 12 | √22 / 4 | 3√2 / 4 | r3 = 5√6 / 12, r6 = √6 / 4 |
7√3 | 23√2 / 12 |
| cuboctahedron, triangular gyrobicupola |
8T+6S | 12 | 1 | √3 / 2 | r3 = √6 / 3, r4 = √2 / 2 |
6 + 2√3 | 5√2 / 3 |
| truncated cube | 8T+6O | 24 | √(7 + 4√2) / 2 | (2 + √2) / 2 | r3 = √(51 + 36√2) / 6, r8 = (1 + √2) / 2 |
2(6 + 6√2 + √3) | 7(3 + 2√2) / 3 |
| truncated octahedron | 6S+8H | 24 | √10 / 2 | 3 / 2 | r4 = √2, r6 = √6 / 2 |
6(1 + 2√3) | 8√2 |
| (small) rhombicuboctahedron | 8T+18S | 24 | √(5 + 2√2) / 2 | √(4 + 2√2) / 2 = 1 / √(2 − √2) |
r3 = (3√3 + √6) / 6, r4 = (1 + √2) / 2 |
18 + 2√3 | (12 + 10√2) / 3 |
| (rhombi)truncated cuboctahedron, great rhombicuboctahedron |
12S+8H +6O |
48 | √(13 + 6√2) / 2 | √(12 + 6√2) / 2 | r4 = (3 + √2) / 2, r6 = (√6 + √3) / 2, r8 = (1 + 2√2) / 2 |
12(2 + √2 + √3) | 22 + 14√2 |
| snub cube | 32T+6S | 24 | √((3 − t) / (2 − t)) / 2 | 1 / (2√(2 − t)) | r3 = √((t + 1) / (6 − 3t)) / 2, r4 = √((1 − t) / (t − 2)) / 2 |
6 + 8√3 | (8t + 6) / (3√(2t2 − 6)) |
| icosidodecahedron | 20T+12P | 30 | (1 + √5) / 2 | √(5 + 2√5) / 2 | r3 = √(42 + 18√5) / 6, r5 = √(25 + 10√5) / 5 |
5√3 + 3√(25 + 10√5) | (45 + 17√5) / 6 |
| truncated dodecahedron | 20T+12D | 60 | √(74 + 30√5) / 4 | (5 + 3√5) / 4 | r3 = √3 × (9 + 5√5) / 12, r10 = √(50 + 22√5) / 4 |
5(√3 + 6√(5 + 2√5)) | 5(99 + 47√5) / 12 |
| truncated icosahedron, “soccer ball” |
12P+20H | 60 | √(58 + 18√5) / 4 | (3 + 3√5) / 4 | r5 = √(1250 + 410√5) / 20, r6 = √(42 + 18√5) / 4 |
30√3 + 3√(25 + 10√5) | (125 + 43√5) / 4 |
| (small) rhombicosidodecahedron | 20T+30S +12P |
60 | √(11 + 4√5) / 2 | √(10 + 4√5) / 2 | r3 = (3√3 + 2√15) / 6, r4 = (2 + √5) / 2, r5 = 3√(25 + 10√5) / 10 |
30 + 5√3 + 3√(25 + 10√5) | (60 + 29√5) / 3 |
| (rhombi)truncated icosidodecahedron, great rhombicosidodecahedron |
30S+20H +12D |
120 | √(31 + 12√5) / 2 | √(30 + 12√5) / 2 | r4 = (3 + 2√5) / 2, r6 = (2√3 + √15) / 2, r10 = √(25 + 10√5) / 2 |
30(1 + √3 + √(5 + 2√5)) | 95 + 50√5 |
| snub dodecahedron | 80T+12P | 60 | √((2 − u) / (1 − u)) / 2 | 1 / (2√(1 − u)) | r3 = vφ√(3(v(v + φ) + 1)) / 6 ≈ 2.07708966, r5 = √(20(5(v + 1 / v)(1 + 2φ) + (12 + 11φ))) / 20 ≈ 1.98091595 |
20√3 + 3√(25 + 10√5) | (5 + 5√5)√(18 + 6√5 + w(3 + 3√5 + w)) / (6√3) + (5 + 3√5)√(72 + (5 + √5)w(3 + 3√5 + w)) / (24√2) ≈ 37.6166500 |
Sources:
- Wikipedia: Archimedean solid
- MathWorld: Archimedean solid
- Wikipedia (de): Abgeschrägtes Dodekaeder
- dmccooey – Archimedean solids
- WolframAlpha
Johnson solids
(many still missing; there are 92 non-uniform Johnson solids in all)
| # | Name | Faces | Vertices | Circumradius | Surface area | Volume |
|---|---|---|---|---|---|---|
| J1 | square pyramid | 4T+1S | 5 | √2 / 2 | 1 + √3 | √2 / 6 |
| J2 | pentagonal pyramid | 5T+1P | 6 | √(10 + 2√5) / 4 | √(10(10 + √5 + √(15(5 + 2√5)))) / 4 | (5 + √5) / 24 |
| J3 | triangular cupola | 4T+3S +1H |
9 | 1 | 3 + 5√3 / 2 | 5√2 / 6 |
| J4 | square cupola | 4T+5S +1O |
12 | √(5 + 2√2) / 2 | 7 + 2√2 + √3 | 1 + 2√2 / 3 |
| J5 | pentagonal cupola | 5T+5S +1P+1D |
15 | √(11 + 4√5) / 2 | (5(4 + √3) + √(5(145 + 62√5))) / 4 | (5 + 4√5) / 6 |
| J6 | pentagonal rotunda | 10T+6P +1D |
20 | (1 + √5) / 2 | (5√3 + √(10(65 + 29√5))) / 2 | (45 + 17√5) / 12 |
| J12 | triangular bipyramid, triangular tegum |
6T | 5 | – | 3√3 / 2 | √2 / 6 |
| J13 | pentagonal bipyramid, pentagonal tegum |
10T | 7 | – | 5√3 / 2 | (5 + √5) / 12 |
| J63 | tridiminished icosahedron | 5T+3P | 9 | √(10 + 2√5) / 4 | (5√3 + 3√(5(5 + 2√5))) / 4 | (15 + 7√5) / 24 |
| J80 | parabidiminished rhombicosidodecahedron |
10T+20S +10P+2D |
50 | √(11 + 4√5) / 2 | 5(8 + √3 + √(85 + 38√5)) / 2 | 5(11 + 5√5) / 3 |
| J84 | snub disphenoid | 12T | 8 | – | 3√3 | ~0.85949 |
| J86 | sphenocorona | 12T+2S | 10 | – | 2 + 3√3 | √(2(2 + 3√6 + 2√(13 + 3√6))) / 4 |
| J88 | sphenomegacorona | 16T+2S | 12 | – | 2 + 4√3 | ~1.948 |
| J89 | hebesphenomegacorona | 18T+3S | 14 | – | 3 + 9√3 / 2 | ~2.9129 |
| J90 | disphenocingulum | 20T+4S | 16 | – | 4 + 5√3 | ~3.7776 |
| J91 | bilunabirotunda | 8T+2S+4P | 14 | – | 2 + 2√3 + √(5(5 + 2√5)) | (17 + 9√5) / 12 |
| J92 | triangular hebesphenorotunda |
13T+3S +3P+1H |
18 | – | 3 + √(6(218 + 15√5 + 19√(15(5 + 2√5)))) / 4 | (15 + 7√5) / 6 |
Inradii:
- triangular bipyramid: √6 / 9
- pentagonal bipyramid: (5√3 + √15) / 30
Sources:
(TODO: add these values to geometrymagic.html)