# Great dodecahedron cone

Great dodecahedron cone
Rank3
Notation
Elements
Faces5 triangles, 5 pentagons
Edges5+5+10
Vertices1+5+5
Vertex figures1 pentagram, edge length (1+5)/2
5 butterflies, edge lengths 1 and (1+5)/2
5 isosceles triangles, edge lengths 1, (1+5)/2, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Dihedral angles3–5 #1: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
5–5: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
3–5 #2: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Related polytopes
ArmyGyepip
Convex hullGyroelongated pentagonal pyramid
Abstract & topological properties
Flag count80
Euler characteristic1
OrientableNo
Genus1
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The great dodecahedron cone, or gadcone, is a nonconvex orbiform polyhedron and an edge-faceting of the icosahedron. Its faces are 5 pentagons and 5 triangles. It is named as such because the 5 pentagons are arranged around one vertex similar to the configuration found in the great dodecahedron.

It is the 5-5-3 acrohedron generated by Green's rules.

## Vertex coordinates

Its vertex coordinates are the same as those of the icosahedron with any 1 vertex removed, which is equivalent to those of the gyroelongated pentagonal pyramid.