Rhombic triacontahedron

3D view of a rhombic triacontahedron ( animation )

Mesh of a rhombic triacontahedron

A rhombic triacontahedron is a Catalan solid and dual to the icosidodecahedron . It is also the envelope that is described by the union of the penetration of a dodecahedron and an icosahedron . A rhombic triacontahedron can also be obtained by placing straight pyramids on an icosahedron or dodecahedron, of which two side faces complement each other to form one.

The rhombic triacontahedron has 30 rhombic faces, 32 corners and 60 edges . 5 edges adjoin 12 of the corners and 3 edges adjoin the remaining 20 corners. The length ratio of the diagonals of the rhombic surfaces corresponds exactly to the golden section .

Related polyhedra

If pyramids with the flank lengths and are placed on the 30 boundary surfaces of the rhombic triacontahedron , a general hexakisicosahedron is created , provided the following condition is met: ${\ displaystyle b}$ ${\ displaystyle c \, (<b)}$

{\ displaystyle {\ frac {a} {10}} {\ sqrt {50 + 10 {\ sqrt {5}}}} \, <b <\, {\ frac {a} {10}} {\ sqrt { 70 + 2 {\ sqrt {5}}}}}

The special hexakisicosahedron with equal surface angles at the edges and is created when is.

{\ displaystyle a}

{\ displaystyle b}

{\ displaystyle b = {\ frac {a} {2}} \, \ left (3 {\ sqrt {5}} - 5 \ right)}

Assumes the aforementioned maximum value, the hexakisicosahedron degenerates into a deltoidal hexacontahedron with the edge lengths and . ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle b}$

Formulas

For the polyhedron

Sizes of a rhombic triacontahedron
volume	${\ displaystyle V = 4a ^ {3} {\ sqrt {5 + 2 {\ sqrt {5}}}}}$
Surface area	${\ displaystyle A_ {O} = 12a ^ {2} {\ sqrt {5}}}$
Inc sphere radius	${\ displaystyle \ rho = a \, {\ sqrt {\ frac {5 + 2 {\ sqrt {5}}} {5}}}}$
Edge ball radius	${\ displaystyle r = \, {\ frac {a} {5}} \ left (5 + {\ sqrt {5}} \ right)}$
Face angle = 144 °	${\ displaystyle \ cos \, \ alpha = - {\ frac {1} {4}} \ left (1 + {\ sqrt {5}} \ right)}$

For the rhombuses

Sizes of the diamonds
Area	${\ displaystyle A = {\ frac {2} {5}} a ^ {2} {\ sqrt {5}}}$
Inscribed radius	${\ displaystyle r = {\ frac {a} {5}} {\ sqrt {5}}}$
Long diagonal	${\ displaystyle e = a \, {\ sqrt {\ frac {10 + 2 {\ sqrt {5}}} {5}}} = {\ frac {f} {2}} \ left (1 + {\ sqrt {5}} \ right)}$
Short diagonal	${\ displaystyle f = a \, {\ sqrt {\ frac {10-2 {\ sqrt {5}}} {5}}}}$
Pointed angle (2) ≈ 63 ° 26 ′ 6 ″	${\ displaystyle \ cos \, \ alpha = {\ frac {1} {5}} {\ sqrt {5}}}$
Obtuse angles (2) ≈ 116 ° 33 ′ 54 ″	${\ displaystyle \ cos \, \ beta = - {\ frac {1} {5}} {\ sqrt {5}}}$