There are five regular polyhedra. These are they: tetrahedron, cube/hexahedron, octahedron, dodecahedron and icosahedron. Here is a short summary in form of a table about the vertices, edges, faces and volumes of these solids:
Tetrahedron 
Cube/ Hexahedron 
Octahedron 
Dodecahedron 
Icosahedron 

v vertices 
4 
8 
6 
20 
12 
e edges 
6 
12 
12 
30 
30 
f faces 
4 
6 
8 
12 
20 
V volume 
√2 /12*a³ 
a³ 
√2 /3*a³ 
(15+7*√5)/4*a³ 
(15+5*√5)/12*a³ 
We could group them into three groups:
• The triangular faced ones {tetrahedron, octahedron, icosahedron}
• The square faced one {cube}
• The pentagon faced one {dodecahedron}
If we put these polygons together like this, we can recognize the golden ratio in it: Here we add a triangle, square and a pentagon together in the following way. Then we connect the point A and H with each other – and exactly where this line crosses the square the golden ratio appears:
Source
Also, if you didn’t know about the pentagram, even there appears the golden ratio:
Source: Wikipedia
And, two of the platonic solids have the golden ratio in themselves: the icosahderon and the dodecahedron.
Imagine 3 planes put together like this:
In a dodechedron the exact same planes can be put into it like this:
But if we want the edges of the planes to be on the edges of the dodecahedron, we have to use another ratio:
Then the ratio is 1 to phi².