Dodecahedronal, Icosahedronal Symmetry: The cosines of the angle between adjacent face normals are cos(∙) = 1 5 and cos(∙) = 5 3 respectively

Dodecahedron

edge length     50 - 22 5            1     12 + 4 5 6    
inscribed      1         25 + 11 5 40         5 + 2 5 15    
center of edge radius     5 - 5 2         3 + 5 4         3 + 5 6    
superscribed     15 - 6 5         9 + 3 5 8            1
pentagon height     15 - 5 5 2     5 + 2 5 2     5 + 5 6    
pentagon area        ∙ 25 + 10 5 4            ∙
total surface area     30 2 65 - 29 5         3 5 5 + 2 5            ∙
volume     10 2 65 - 29 5         5 47 + 21 5 2 2            ∙

wiki/Exact_trigonometric_constants#Uses_for_cnstants Volume, where a is the length of an edge

15 + 7 5 4 a 3

sin π /5 = 5 - 5 8

Divide the circle into 120 divisions wthout transendental formulas 16 sin π /60 = 2 1 - 3 5 + 5 + 2 5 - 1 3 + 1

For pi/7 angles the solution is transcendental but for pi/60 the solution is irrational