Take the tour ×  Could someone describe to me how to find the
angle between two intersecting pentagonal faces on a great
dodecahedron? Thanks  share|improve this question Doesn't a
great dodecahedron have triangular faces? A regular dodecahedron
has pentagonal faces... –  mjqxxxx Dec 2 '11 at 4:01 There is an
illustrative discussion of a comment similar to that of @mjqxxxx
above in Proofs and refutations by Imre Lakatos on pp. 16--17
and pp. 30--33. –  Robert Haraway Dec 2 '11 at 6:13 add comment
1 Answer  The answer is $$\arccos \frac{1}{\sqrt 5} \approx
63^\circ 26' 06'' \approx 1.107148718 \; \mbox{radians} $$ and
thereby precisely supplementary to the dihedral angles of the
ordinary dodecahedron.  The three books I have on this sort of
thing are Platonic and Archimedean Solids by Daud Sutton,
Shapes, Space, and Symmetry by Alan Holden, and Regular
Polytopes by H. S. M. Coxeter. Sutton gives angles in degree-
minute-second format, while Coxeter shows how to find them on
pages 20-22.   share|improve this an