Large dodecahedron

The (also, especially Austrian : the) large dodecahedron is a regular polyhedron and belongs to the Kepler star bodies ; it is bounded by 60 isosceles triangles .

Emergence

The basic body is the icosahedron . The great dodecahedron is the result of twelve mutually intersecting regular pentagons that can be found in the icosahedron (see cut surfaces of an icosahedron ); this star body is quasi a reduced, "scratched out" icosahedron, whereby the 20 cutouts have the shape of triangular pyramids.

Formulas

Sizes in the cut out tetrahedron

Sizes of a large dodecahedron with edge length a
volume	${\ displaystyle V = {\ frac {5} {4}} a ^ {3} \ left ({\ sqrt {5}} - 1 \ right)}$
Surface area	${\ displaystyle A_ {O} = 15a ^ {2} {\ sqrt {5-2 {\ sqrt {5}}}}}$
Umkugelradius	${\ displaystyle R = {\ frac {a} {4}} {\ sqrt {10 + 2 {\ sqrt {5}}}}}$
Pyramid height	${\ displaystyle k = {\ frac {a} {6}} {\ sqrt {3}} \ left (3 - {\ sqrt {5}} \ right)}$
Ridge length	${\ displaystyle s = {\ frac {a} {2}} \ left ({\ sqrt {5}} - 1 \ right)}$
Face angle (outside) ≈ 116 ° 33 ′ 54 ″	${\ displaystyle \ cos \, \ alpha _ {1} = - {\ frac {1} {5}} {\ sqrt {5}}}$
Face angle (inside) ≈ 63 ° 26 ′ 6 ″	${\ displaystyle \ cos \, \ alpha _ {2} = {\ frac {1} {5}} {\ sqrt {5}}}$

Remarks

↑ Let the sides of the triangle be denoted by a (side of the icosahedron) and s (leg).
↑ ^a ^b The edge length of the circumscribed icosahedron be with a designated.

Web links

Commons : Large Dodecahedron - collection of images, videos and audio files

Eric W. Weisstein : Large dodecahedron . In: MathWorld (English).

[1] Let the sides of the triangle be denoted by a (side of the icosahedron) and s (leg).

[iko-2] The edge length of the circumscribed icosahedron be with a designated.