That Other Dodecahedron

Geometry Level 5

We're all familiar with the regular dodecahedron that has 12 identical flat regular pentagonal faces. If the edge length is 1, then the (area)² of each of those 12 faces is \[\frac { 25+10\sqrt { 5 } }{ 16 } \] But there is another dodecahedron that also has 12 identical flat pentagonal faces. If the edge length is 1 also, what is the (area)² of each of those 12 faces? If the answer is expressed as: \[\frac { a+b\sqrt { c } }{ d }\] where a, b, c, d are irreducible positive integers, find \(a+b+c+d\)
Those "12 identical flat pentagonal faces" need not be regular, but they must not intersect.

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