Chamfered dodecahedron

Chamfered dodecahedron
Rank	3
Type	Near-miss
Notation
Bowers style acronym	Chado
Conway notation	cD ; t5daD
Elements
Faces	12 pentagons, 30 rectangular-symmetric hexagons
Edges	60+60
Vertices	20+60
Vertex figures	20 equilateral triangles
	60 isosceles triangles
Measures (edge length 1)
Dihedral angles	5-6:
	6-6: 144°
Central density	1
Number of external pieces	42
Level of complexity	4
Related polytopes
Army	Chamfered dodecahedron
Regiment	Chamfered dodecahedron
Dual	Pentakis icosidodecahedron
Conjugate	None
Abstract & topological properties
Flag count	480
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Flag orbits	4
Convex	Yes
Nature	Tame

The chamfered dodecahedron (OBSA: chado), also known as the order-5 truncated rhombic triacontahedron, is one of the near-miss Johnson solids. It has 12 pentagons and 30 hexagons as faces, and 80 order-3 vertices divided into two sets of 20 and 60 each. It can be made equilateral, but the hexagons have two angle sizes.

The hexagonal faces have angles of $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$ on one pair of opposite vertices, and angles of $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$ on the four remaining vertices.

The canonical variant with midradius 1 has two edge lengths: one of length ${\frac {{\sqrt {5}}-3+2{\sqrt {5-2{\sqrt {5}}}}}{2}}\approx 0.34458$ and the other of length ${\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 0.36327$ with the same dihedral angles as the equilateral variant.