Chamfered dodecahedron

The chamfered dodecahedron, also known as the order-5 truncated rhombic dodecahedron, 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 $$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$$ on one pair of opposite vertices, and angles of $$\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747^\circ$$ on the four remaining vertices.

The canonical variant with midradius 1 has two edge lengths: one of length $$\frac{\sqrt5-3+2\sqrt{5-2\sqrt5}}{2} ≈ 0.34458$$ and the other of length $$\frac{\sqrt{5-2\sqrt5}}{2} ≈ 0.36327$$ with the same dihedral angles as the equilateral variant.

It can also be viewed as an order-5-truncated rhombic triacontahedron, or as an icosahedrally-symmetric Goldberg polyhedron of index (2,0).

It is the convex core of the uniform rhombidodecadodecahedron.