Tables for the Regular Polyhedra

For quite some time now, I have been looking in books and online for a set of tables with formulae for conversion between various measures of the platonic solids (the regular polyhedra). None quite fitted my requirements, and so I created my own.

My requirements included:

  • The formulae should all be of a similar form.
  • Where there is a change of dimension, formulae should be given both in terms of the source and the target dimensons.
  • No formula should have a surd (root) in the denominator.
  • The terms in a surd should have reduced factors. (So, in particular, any integer under a square root should be square-free.)

I use the symbols:

a = length of an edge
r = inradius, i.e. the radius of the insphere
ρ (rho) = midradius, i.e. the radius of the midsphere
R = circumradius, i.e. the radius of the circumsphere
S = total surface area of polyhedron
C = volume or capacity of polyhedron

The tables were generated with the help of Maxima and Python. Much of the work to produce the final forms required hand calculation. The final tables have been checked by computer for correctness.

It is possible that these formulae are not the simplest possible, if I have missed an available simplification.

For example, it happens that:

(a + b√c)² = h + k·√c

where

h = a² + b²c
k = 2ab

so

h + k·√c = (a + b√c)²

where, solving for a and b,

a = ±√( (h ± √(h²−k²c)) / 2 )
b = ±√( (h ± √(h²−k²c)) / 2c )

and so it is possible that a form such as

√( h + k·√c )

might simplify to

a + b√c

for certain integers c, h and k.

My formulae include consideration for the following spheres:

  • The insphere is the largest that fits inside the polyhedron; it touches the centres of the faces.
  • The midsphere passes through the midpoints of the edges.
  • The circumsphere is the smallest that fits around the polyhedron; it passes through the vertices.

The formulae are for the radius of each of these spheres. I do not calculate the volumes or surface areas of these spheres, as those are related directly to the radii by the formulae

Area = 2τ rad²
Vol = 2τ rad³ / 3

and

rad = √( Area / 2τ )
rad = ∛( 3 Vol / 2τ )

(where π=τ/2)

In addition, I do not calculate the areas of individual faces. Those are easily obtained by dividing the total surface area S by the number of faces.

The tables are also available in PDF format.

 

given a
= edge
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
a a a a a
r =
inradius
a·√6/12
≈ 0.204124⁺ · a
a·1/2
≈ 0.5 · a
a·√6/6
≈ 0.408248⁺ · a
a·√(10·(25+11·√5))/20
≈ 1.113516⁺ · a
a·√(6·(7+3·√5))/12
≈ 0.755761⁺ · a
ρ =
midradius
a·√2/4
≈ 0.353553⁺ · a
a·√2/2
≈ 0.707107¯ · a
a·1/2
≈ 0.5 · a
a·(3+√5)/4
≈ 1.309017¯ · a
a·(1+√5)/4
≈ 0.809017¯ · a
R =
circumradius
a·√6/4
≈ 0.612372⁺ · a
a·√3/2
≈ 0.866025⁺ · a
a·√2/2
≈ 0.707107¯ · a
a·√(6·(3+√5))/4
≈ 1.401259¯ · a
a·√(2·(5+√5))/4
≈ 0.951057¯ · a
S =
surface area
a²·√3
≈ 1.732051¯ · a²
(a·∜3)²
≈ (1.316074⁺ · a)²
a²·6
≈ 6.0 · a²
(a·√6)²
≈ (2.449490¯ · a)²
a²·2·√3
≈ 3.464102¯ · a²
(a·∜12)²
≈ (1.861210¯ · a)²
a²·3·√(5·(5+2·√5))
≈ 20.645729¯ · a²
(a·∜(45·(5+2·√5)))²
≈ (4.543757⁺ · a)²
a²·5·√3
≈ 8.660254⁺ · a²
(a·∜75)²
≈ (2.942831¯ · a)²
C =
volume
a³·√2/12
≈ 0.117851⁺ · a³
(a·⁶√41472/12)³
≈ (0.490280⁺ · a)³
a³·√2/3
≈ 0.471405¯ · a³
(a·⁶√162/3)³
≈ (0.778272¯ · a)³
a³·(15+7·√5)/4
≈ 7.663119¯ · a³
(a·∛(16·(15+7·√5))/4)³
≈ (1.971523⁺ · a)³
a³·5·(3+√5)/12
≈ 2.181695¯ · a³
(a·∛(720·(3+√5))/12)³
≈ (1.296974⁺ · a)³

 

given r
= inradius
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
r·2·√6
≈ 4.898979⁺ · r
r·2
≈ 2.0 · r
r·√6
≈ 2.449490¯ · r
r·√(2·(25−11·√5))
≈ 0.898056¯ · r
r·√(6·(7−3·√5))
≈ 1.323169⁺ · r
r =
inradius
r r r r r
ρ =
midradius
r·√3
≈ 1.732051¯ · r
r·√2
≈ 1.414214¯ · r
r·√6/2
≈ 1.224745¯ · r
r·√(2·(5−√5))/2
≈ 1.175571¯ · r
r·√(6·(3−√5))/2
≈ 1.070466⁺ · r
R =
circumradius
r·3
≈ 3.0 · r
r·√3
≈ 1.732051¯ · r
r·√3
≈ 1.732051¯ · r
r·√(3·(5−2·√5))
≈ 1.258409¯ · r
r·√(3·(5−2·√5))
≈ 1.258409¯ · r
S =
surface area
r²·24·√3
≈ 41.569219⁺ · r²
(r·2·∜108)²
≈ (6.447420¯ · r)²
r²·24
≈ 24.0 · r²
(r·2·√6)²
≈ (4.898979⁺ · r)²
r²·12·√3
≈ 20.784610¯ · r²
(r·2·∜27)²
≈ (4.559014⁺ · r)²
r²·30·√(2·(65−29·√5))
≈ 16.650873⁺ · r²
(r·∜(1800·(65−29·√5)))²
≈ (4.080548⁺ · r)²
r²·30·√(6·(47−21·√5))
≈ 15.162168⁺ · r²
(r·∜(5400·(47−21·√5)))²
≈ (3.893863¯ · r)²
C =
volume
r³·8·√3
≈ 13.856406⁺ · r³
(r·⁶√192)³
≈ (2.401874¯ · r)³
r³·8
≈ 8.0 · r³
(r·2)³
≈ (2.0 · r)³
r³·4·√3
≈ 6.928203⁺ · r³
(r·⁶√48)³
≈ (1.906369¯ · r)³
r³·10·√(2·(65−29·√5))
≈ 5.550291⁺ · r³
(r·⁶√(200·(65−29·√5)))³
≈ (1.770538¯ · r)³
r³·10·√(6·(47−21·√5))
≈ 5.054056⁺ · r³
(r·⁶√(600·(47−21·√5)))³
≈ (1.716116⁺ · r)³

 

given ρ
= midradius
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
ρ·2·√2
≈ 2.828427⁺ · ρ
ρ·√2
≈ 1.414214¯ · ρ
ρ·2
≈ 2.0 · ρ
ρ·(3−√5)
≈ 0.763932⁺ · ρ
ρ·(√5−1)
≈ 1.236068¯ · ρ
r =
inradius
ρ·√3/3
≈ 0.577350⁺ · ρ
ρ·√2/2
≈ 0.707107¯ · ρ
ρ·√6/3
≈ 0.816497¯ · ρ
ρ·√(10·(5+√5))/10
≈ 0.850651¯ · ρ
ρ·√(6·(3+√5))/6
≈ 0.934172⁺ · ρ
ρ =
midradius
ρ ρ ρ ρ ρ
R =
circumradius
ρ·√3
≈ 1.732051¯ · ρ
ρ·√6/2
≈ 1.224745¯ · ρ
ρ·√2
≈ 1.414214¯ · ρ
ρ·√(6·(3−√5))/2
≈ 1.070466⁺ · ρ
ρ·√(2·(5−√5))/2
≈ 1.175571¯ · ρ
S =
surface area
ρ²·8·√3
≈ 13.856406⁺ · ρ²
(ρ·2·∜12)²
≈ (3.722419⁺ · ρ)²
ρ²·12
≈ 12.0 · ρ²
(ρ·2·√3)²
≈ (3.464102¯ · ρ)²
ρ²·8·√3
≈ 13.856406⁺ · ρ²
(ρ·2·∜12)²
≈ (3.722419⁺ · ρ)²
ρ²·6·√(10·(25−11·√5))
≈ 12.048685¯ · ρ²
(ρ·∜(360·(25−11·√5)))²
≈ (3.471122¯ · ρ)²
ρ²·10·√(6·(7−3·√5))
≈ 13.231691¯ · ρ²
(ρ·∜(600·(7−3·√5)))²
≈ (3.637539⁺ · ρ)²
C =
volume
ρ³·8/3
≈ 2.666667¯ · ρ³
(ρ·2·∛9/3)³
≈ (1.386723¯ · ρ)³
ρ³·2·√2
≈ 2.828427⁺ · ρ³
(ρ·√2)³
≈ (1.414214¯ · ρ)³
ρ³·8·√2/3
≈ 3.771236⁺ · ρ³
(ρ·2·⁶√162/3)³
≈ (1.556543⁺ · ρ)³
ρ³·2·(3·√5−5)
≈ 3.416408¯ · ρ³
(ρ·∛(2·(3·√5−5)))³
≈ (1.506110¯ · ρ)³
ρ³·10·(√5−1)/3
≈ 4.120227¯ · ρ³
(ρ·∛(90·(√5−1))/3)³
≈ (1.603148⁺ · ρ)³

 

given R
= circumradius
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
R·2·√6/3
≈ 1.632993⁺ · R
R·2·√3/3
≈ 1.154701¯ · R
R·√2
≈ 1.414214¯ · R
R·√(6·(3−√5))/3
≈ 0.713644⁺ · R
R·√(10·(5−√5))/5
≈ 1.051462⁺ · R
r =
inradius
R·1/3
≈ 0.333333⁺ · R
R·√3/3
≈ 0.577350⁺ · R
R·√3/3
≈ 0.577350⁺ · R
R·√(15·(5+2·√5))/15
≈ 0.794654⁺ · R
R·√(15·(5+2·√5))/15
≈ 0.794654⁺ · R
ρ =
midradius
R·√3/3
≈ 0.577350⁺ · R
R·√6/3
≈ 0.816497¯ · R
R·√2/2
≈ 0.707107¯ · R
R·√(6·(3+√5))/6
≈ 0.934172⁺ · R
R·√(10·(5+√5))/10
≈ 0.850651¯ · R
R =
circumradius
R R R R R
S =
surface area
R²·8·√3/3
≈ 4.618802⁺ · R²
(R·2·∜108/3)²
≈ (2.149140¯ · R)²
R²·8
≈ 8.0 · R²
(R·2·√2)²
≈ (2.828427⁺ · R)²
R²·4·√3
≈ 6.928203⁺ · R²
(R·2·∜3)²
≈ (2.632148⁺ · R)²
R²·2·√(10·(5−√5))
≈ 10.514622⁺ · R²
(R·∜(40·(5−√5)))²
≈ (3.242626¯ · R)²
R²·2·√(30·(3−√5))
≈ 9.574541⁺ · R²
(R·∜(120·(3−√5)))²
≈ (3.094276¯ · R)²
C =
volume
R³·8·√3/27
≈ 0.513200⁺ · R³
(R·2·⁶√3/3)³
≈ (0.800625¯ · R)³
R³·8·√3/9
≈ 1.539601¯ · R³
(R·2·⁶√27/3)³
≈ (1.154701¯ · R)³
R³·4/3
≈ 1.333333⁺ · R³
(R·∛36/3)³
≈ (1.100642⁺ · R)³
R³·2·√(30·(3+√5))/9
≈ 2.785164¯ · R³
(R·⁶√(1080·(3+√5))/3)³
≈ (1.406966¯ · R)³
R³·2·√(2·(5+√5))/3
≈ 2.536151¯ · R³
(R·⁶√(648·(5+√5))/3)³
≈ (1.363719⁺ · R)³

 

given S
= surface area
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
√S·∜27/3
≈ 0.759836¯ · √S
√(S·√3/3)
≈ √(0.577350⁺ · S)
√S·√6/6
≈ 0.408248⁺ · √S
√(S·1/6)
≈ √(0.166667¯ · S)
√S·∜1728/12
≈ 0.537285¯ · √S
√(S·√3/6)
≈ √(0.288675⁺ · S)
√S·∜(225·(5−2·√5))/15
≈ 0.220082⁺ · √S
√(S·√(5−2·√5)/15)
≈ √(0.048436⁺ · S)
√S·∜675/15
≈ 0.339809¯ · √S
√(S·√3/15)
≈ √(0.115470⁺ · S)
r =
inradius
√S·∜12/12
≈ 0.155101¯ · √S
√(S·√3/72)
≈ √(0.024056⁺ · S)
√S·√6/12
≈ 0.204124⁺ · √S
√(S·1/24)
≈ √(0.041667¯ · S)
√S·∜3/6
≈ 0.219346¯ · √S
√(S·√3/36)
≈ √(0.048113¯ · S)
√S·∜(360·(65+29·√5))/60
≈ 0.245065⁺ · √S
√(S·√(10·(65+29·√5))/600)
≈ √(0.060057¯ · S)
√S·∜(600·(47+21·√5))/60
≈ 0.256814⁺ · √S
√(S·√(6·(47+21·√5))/360)
≈ √(0.065954¯ · S)
ρ =
midradius
√S·∜108/12
≈ 0.268642⁺ · √S
√(S·√3/24)
≈ √(0.072169¯ · S)
√S·√12/12
≈ 0.288675⁺ · √S
√(S·1/12)
≈ √(0.083333⁺ · S)
√S·∜108/12
≈ 0.268642⁺ · √S
√(S·√3/24)
≈ √(0.072169¯ · S)
√S·∜(1800·(25+11·√5))/60
≈ 0.288091⁺ · √S
√(S·√(2·(25+11·√5))/120)
≈ √(0.082997¯ · S)
√S·∜(5400·(7+3·√5))/60
≈ 0.274911⁺ · √S
√(S·√(6·(7+3·√5))/120)
≈ √(0.075576⁺ · S)
R =
circumradius
√S·∜12/4
≈ 0.465302⁺ · √S
√(S·√3/8)
≈ √(0.216506⁺ · S)
√S·√2/4
≈ 0.353553⁺ · √S
√(S·1/8)
≈ √(0.125 · S)
√S·∜27/6
≈ 0.379918¯ · √S
√(S·√3/12)
≈ √(0.144338¯ · S)
√S·∜(200·(5+√5))/20
≈ 0.308392⁺ · √S
√(S·√(2·(5+√5))/40)
≈ √(0.095106¯ · S)
√S·∜(27000·(3+√5))/60
≈ 0.323177⁺ · √S
√(S·√(30·(3+√5))/120)
≈ √(0.104444¯ · S)
S =
surface area
S S S S S
C =
volume
√S³·∜12/36
≈ 0.051700⁺ · √S³
√(S·⁶√243/18)³
≈ √(0.138781¯ · S)³
√S³·√6/36
≈ 0.068041⁺ · √S³
√(S·1/6)³
≈ √(0.166667¯ · S)³
√S³·∜3/18
≈ 0.073115⁺ · √S³
√(S·⁶√972/18)³
≈ √(0.174853¯ · S)³
√S³·∜(360·(65+29·√5))/180
≈ 0.081688⁺ · √S³
√(S·⁶√(250·(65+29·√5))/30)³
≈ √(0.188267¯ · S)³
√S³·∜(600·(47+21·√5))/180
≈ 0.085605¯ · √S³
√(S·⁶√(303750·(47+21·√5))/90)³
≈ √(0.194237⁺ · S)³

 

given C
= volume
tetrahedron
(4)
cube /
hexahedron
(6)
octahedron
(8)
dodecahedron
(12)
icosahedron
(20)
a =
edge
∛C·⁶√72
≈ 2.039649¯ · ∛C
∛(C·6·√2)
≈ ∛(8.485281⁺ · C)
∛C ∛C·⁶√288/2
≈ 1.284898⁺ · ∛C
∛(C·3·√2/2)
≈ ∛(2.121320⁺ · C)
∛C·∛(25·(7·√5−15))/5
≈ 0.507222⁺ · ∛C
∛(C·(7·√5−15)/5)
≈ ∛(0.130495⁺ · C)
∛C·∛(75·(3−√5))/5
≈ 0.771025⁺ · ∛C
∛(C·3·(3−√5)/5)
≈ ∛(0.458359⁺ · C)
r =
inradius
∛C·⁶√243/6
≈ 0.416342¯ · ∛C
∛(C·√3/24)
≈ ∛(0.072169¯ · C)
∛C·1/2
≈ 0.5 · ∛C
∛(C·1/8)
≈ ∛(0.125 · C)
∛C·⁶√972/6
≈ 0.524558¯ · ∛C
∛(C·√3/12)
≈ ∛(0.144338¯ · C)
∛C·⁶√(250·(65+29·√5))/10
≈ 0.564800⁺ · ∛C
∛(C·√(10·(65+29·√5))/200)
≈ ∛(0.180171¯ · C)
∛C·⁶√(303750·(47+21·√5))/30
≈ 0.582711⁺ · ∛C
∛(C·√(6·(47+21·√5))/120)
≈ ∛(0.197861¯ · C)
ρ =
midradius
∛C·∛3/2
≈ 0.721125¯ · ∛C
∛(C·3/8)
≈ ∛(0.375 · C)
∛C·√2/2
≈ 0.707107¯ · ∛C
∛(C·√2/4)
≈ ∛(0.353553⁺ · C)
∛C·⁶√288/4
≈ 0.642449⁺ · ∛C
∛(C·3·√2/16)
≈ ∛(0.265165⁺ · C)
∛C·∛(25·(5+3·√5))/10
≈ 0.663962⁺ · ∛C
∛(C·(5+3·√5)/40)
≈ ∛(0.292705⁺ · C)
∛C·∛(75·(1+√5))/10
≈ 0.623773¯ · ∛C
∛(C·3·(1+√5)/40)
≈ ∛(0.242705⁺ · C)
R =
circumradius
∛C·⁶√15552/4
≈ 1.249025¯ · ∛C
∛(C·9·√3/8)
≈ ∛(1.948557⁺ · C)
∛C·√3/2
≈ 0.866025⁺ · ∛C
∛(C·3·√3/8)
≈ ∛(0.649519⁺ · C)
∛C·∛6/2
≈ 0.908560⁺ · ∛C
∛(C·3/4)
≈ ∛(0.75 · C)
∛C·⁶√(168750·(3−√5))/10
≈ 0.710749⁺ · ∛C
∛(C·3·√(30·(3−√5))/40)
≈ ∛(0.359045⁺ · C)
∛C·⁶√(56250·(5−√5))/10
≈ 0.733289¯ · ∛C
∛(C·3·√(10·(5−√5))/40)
≈ ∛(0.394298⁺ · C)
S =
surface area
∛C²·6·⁶√3
≈ 7.205622¯ · ∛C²
∛(C·6·∜108)²
≈ ∛(19.342259¯ · C)²
∛C²·6
≈ 6.0 · ∛C²
∛(C·6·√6)²
≈ ∛(14.696938⁺ · C)²
∛C²·⁶√34992
≈ 5.719106¯ · ∛C²
∛(C·6·∜27)²
≈ ∛(13.677042⁺ · C)²
∛C²·3·⁶√(200·(65−29·√5))
≈ 5.311614¯ · ∛C²
∛(C·3·∜(1800·(65−29·√5)))²
≈ ∛(12.241644⁺ · C)²
∛C²·3·⁶√(600·(47−21·√5))
≈ 5.148349¯ · ∛C²
∛(C·3·∜(5400·(47−21·√5)))²
≈ ∛(11.681589¯ · C)²
C =
volume
C C C C C
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