Pentakis dodecahedron

The pentakis dodecahedron, or pakid, is one of the 13 Catalan solids. It has 60 isosceles triangles as faces, with 12 order-5 and 20 order-6 vertices. It is the dual of the uniform truncated icosahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron, where the edges of the icosahedron are $$3\frac{7+5\sqrt5}{38} ≈ 1.43529$$ times the length of those of the dodecahedron. Using an icosahedron that is any number less than $$\frac{1+\sqrt5}{2} ≈ 1.61803$$ times the edge length of the dodecahedron including if the two have the same edge length) gives a fully symmetric variant of this polyhedron.

Each face of this polyhedron is an isosceles triangle with base side length $$\frac{9-\sqrt5}{6} ≈ 1.12732$$ times those of the side edges. These triangles have apex angle $$\arccos\left(\frac{9\sqrt5-7}{36}\right) ≈ 68.61873°$$ and base angles $$\arccos\left(\frac{9-\sqrt5}{12}\right) ≈ 55.69064°$$.