In[39]:= PolyhedronData["PentagonalCupola", "SurfaceArea"] Out[39]= 1/4 (20 + Sqrt[10 (80 + 31 Sqrt[5] + Sqrt[2175 + 930 Sqrt[5]])]) (*Depth 3. Probably lifted from http://en.wikipedia.org/wiki/Pentagonal_cupola*) In[40]:= Expand[RationalizeDenominator[Strad[%]]] Out[40]= 5 + (5 Sqrt[3])/4 + 1/4 5^(3/4) Sqrt[62 + 29 Sqrt[5]] (*Depth 2*) In[41]:= MinimalPolynomial[% - %%] Out[41]= #1 & Similarly for PolyhedronData["PentagonalGyrobicupola", "SurfaceArea"] (the dual of the (disappointingly squat) rhombic icosahedron) Something I didn't know: "The rhombic triacontahedron is also interesting in that its vertices include the arrangement of all the platonic solids <http://en.wikipedia.org/wiki/Platonic_solids>. It contains ten tetrahedrons <http://en.wikipedia.org/wiki/Tetrahedron>, five hexahedrons <http://en.wikipedia.org/wiki/Cube>, five octahedrons <http://en.wikipedia.org/wiki/Octahedron>, an icosahedron <http://en.wikipedia.org/wiki/Icosahedron> and a dodecahedron <http://en.wikipedia.org/wiki/Dodecahedron>." --rwg