Surface area of the Platonic solids
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Numbers
constraint
solid
area
$a = 1$
tetrahedron:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248575
$a = 1$
cube:
6
$a = 1$
octahedron:
3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151
$a = 1$
dodecahedron:
20.64572880706760307310814372866331519288849004012237995048513648428642790650759477598929489665105288
$a = 1$
icosahedron:
8.660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878
$r = 1$
tetrahedron:
41.56921938165305504465871219614093680662732609144913507333936750684639240581120088994750848217396581
$r = 1$
cube:
24
$r = 1$
octahedron:
20.78460969082652752232935609807046840331366304572456753666968375342319620290560044497375424108698290
$r = 1$
dodecahedron:
16.65087308554653080721129634098551772221279463864749660133526159061651012199973570944881668240785065
$r = 1$
icosahedron:
15.16216843079671197462984573581723208302614982393675749883782673469245527651989477030483530174158427
$\rho = 1$
tetrahedron:
13.85640646055101834821957073204697893554244203048304502444645583561546413527040029664916949405798860
$\rho = 1$
cube:
12
$\rho = 1$
octahedron:
13.85640646055101834821957073204697893554244203048304502444645583561546413527040029664916949405798860
$\rho = 1$
dodecahedron:
12.04868495317363679218786035796452853983566764331200087326784824627229472654814420293669360307525122
$\rho = 1$
icosahedron:
13.23169076499214995403073624735217489995494056139551057579847172242315958789421077724151183413072209
$R = 1$
tetrahedron:
4.618802153517006116073190244015659645180814010161015008148818611871821378423466765549723164685996201
$R = 1$
cube:
8
$R = 1$
octahedron:
6.928203230275509174109785366023489467771221015241522512223227917807732067635200148324584747028994302
$R = 1$
dodecahedron:
10.51462224238267212051338169695753214570995864486683563057871046482422292806428036743265257663105141
$R = 1$
icosahedron:
9.574541383273939164915932615493924447762209127520624627382922262292363985214042303772887113855138720
$V = 1$
tetrahedron:
7.205621731056016360052792324097257077790444509355893350110228342695233624114567516268450730218521578
$V = 1$
cube:
6
$V = 1$
octahedron:
5.719105757981619442544453972396562946637442567902081239655857241552507174386170248041811430392081677
$V = 1$
dodecahedron:
5.311613997069083669796666701461086337809888399341493422663761016884993104265681047701440824017902919
$V = 1$
icosahedron:
5.148348556199515646330812946116019064100864116386724148450713675398032479050857713029837325629943109
Definition
This table lists the surface areas of the Platonic solids [1] given that either the edge length $a$ equals $1$, the inner radius $r$ equals $1$, the midradius $\rho$ equals $1$, the outer radius $R$ equals $1$, or the volume $V$ equals $1$.
Parameters
constraint
—   length that is constrained to be $1$
solid
—   Platonic solid