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:
- 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 |
|---|---|---|---|---|---|---|---|
| tetrahedron, disphenoid, digonal antiprism |
4 | 4 | √6 / 4 | √2 / 4 | √6 / 12 | √3 | √2 / 12 |
| cube | 6 | 8 | √3 / 2 | √2 / 2 | 1 / 2 | 6 | 1 |
| octahedron | 8 | 6 | √2 / 2 | 1 / 2 | √6 / 6 | 2√3 | √2 / 3 |
| dodecahedron | 12 | 20 | (√3 + √15) / 4 | (3 + √5) / 4 | √(250 + 110√5) / 20 | 3√(25 + 10√5) | (15 + 7√5) / 4 |
| icosahedron | 20 | 12 | √(10 + 2√5) / 4 | (1 + √5) / 4 | (3√3 + √15) / 12 | 5√3 | (15 + 5√5) / 12 |
Sources:
- MathWorld: regular tetrahedron, regular octahedron, cube, regular dodecahedron, regular icosahedron
- WolframAlpha
Archimedean solids
13 solids. Six types of faces: triangle, square, pentagon, hexagon, octagon, decagon. 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.
Constants used in the table:
- the golden ratio: φ = (1 + √5) / 2
- the tribonacci constant: t = (1 + ∛(19 + 3√33) + ∛(19 − 3√33)) / 3 ≈ 1.83928676
- used in snub dodecahedron: ξ = real zero of x3 + 2x2 − φ2 = (∛(44 + 12φ(9 + √(81φ − 15))) + ∛(44 + 12φ(9 − √(81φ − 15))) − 4) / 6 ≈ 0.943151259
- used in snub dodecahedron: kSD = ∛(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 | 8 | 12 | 7√3 | 23√2 / 12 | |||
| cuboctahedron, triangular gyrobicupola |
14 | 12 | 1 | 6 + 2√3 | 5√2 / 3 | ||
| truncated cube | 14 | 24 | 2(6 + 6√2 + √3) | 7(3 + 2√2) / 3 | |||
| truncated octahedron | 14 | 24 | 6(1 + 2√3) | 8√2 | |||
| rhombicuboctahedron, small rhombicuboctahedron |
26 | 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 |
| truncated cuboctahedron | 26 | 48 | 12(2 + √2 + √3) | 22 + 14√2 | |||
| snub cube | 38 | 24 | 6 + 8√3 | (8t + 6) / (3√(2t2 − 6)) | |||
| icosidodecahedron | 32 | 30 | (1 + √5) / 2 | 5√3 + 3√(25 + 10√5) | (45 + 17√5) / 6 | ||
| truncated dodecahedron | 32 | 60 | 5(√3 + 6√(5 + 2√5)) | 5(99 + 47√5) / 12 | |||
| truncated icosahedron, “soccer ball” |
32 | 60 | √(58 + 18√5) / 4 | r5 = √(1250 + 410√5) / 20, r6 = √(42 + 18√5) / 4 |
30√3 + 3√(25 + 10√5) | (125 + 43√5) / 4 | |
| rhombicosidodecahedron | 62 | 60 | √(11 + 4√5) / 2 | 30 + 5√3 + 3√(25 + 10√5) | (60 + 29√5) / 3 | ||
| truncated icosidodecahedron | 62 | 120 | 30(1 + √3 + √(5 + 2√5)) | 95 + 50√5 | |||
| snub dodecahedron | 92 | 60 | √((2 − ξ) / (1 − ξ)) / 2 | 1 / (2√(1 − ξ)) | 20√3 + 3√(25 + 10√5) | (5 + 5√5)√(18 + 6√5 + kSD(3 + 3√5 + kSD)) / (6√3) + (5 + 3√5)√(72 + (5 + √5)kSD(3 + 3√5 + kSD)) / (24√2) ≈ 37.6166500 |
Sources:
- Wikipedia: Archimedean solid
- Wikipedia (de): Abgeschrägtes Dodekaeder
- MathWorld: small rhombicuboctahedron
- dmccooey – Archimedean solids
- WolframAlpha
Johnson solids
(many still missing)
| # | Name | Faces | Vertices | Circumradius | Midradius | Inradius | Surface area | Volume |
|---|---|---|---|---|---|---|---|---|
| J6 | pentagonal rotunda | 17 | 20 | (45 + 17√5) / 12 | ||||
| J63 | tridiminished icosahedron | 8 | 9 | (15 + 7√5) / 24 |