PolyhedronData@{"DodecahedronStellation", 1} has 12 pentagonal pyramid vertices (solid angle ArcTan[38/41]) and 20 monkey saddles (solid angle 2π, exactly self-complementary <http://gosper.org/saddlekiss.png>). Unfortunately, for these the new function PolyhedronAngle gets 0.) —rwg The Regular Dodecahedron vertex solid angle is π - ArcTan[2/11] == 2 ArcTan[GoldenRatio^5] Almost trivial Cheesewedge Theorem: The dihedral angle of a cheesewedge (in radians) equals the solid angle of the sharp corners at either end of the dihedral edge (in steradians). Unless it's a mild cheese. In[1825]:= solidAngles[π/2, π/2, a] Out[1825]= {2 ArcTan[Sqrt[1 - Cos[a]^2]/(1 + Cos[a])], ArcCos[Cos[a]], 2 ArcCos[1/2 (1 + Cos[a]) Sec[a/2]], 2 ArcSin[(Sqrt[1 - Cos[2 a]] Sec[a/2])/(2 Sqrt[2])], 4 ArcTan[Sqrt[Tan[a/4]^2 Tan[1/4 (-a + \[Pi])] Tan[(a + \[Pi])/4]]]} In[1830]:= Assuming[0 < a < π, FullSimplify@%1825] Out[1830]= {a, a, a, a, a}