Johnson Solids (Page 1)

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(J20) Elongated Pentagonal Cupola
Vertices:  25  (10[3] + {10 + 5}[4])
Faces:22  (5 equilateral triangles + {5 * 3} squares
    + 1 regular pentagon + 1 regular decagon)
Edges:45
Symmetry:  5-fold Pyramidal  (C5v)
(values below based on edge length = 1)
Volume:(5+4*sqrt(5)+15*sqrt(5    
    +2*sqrt(5)))/6    
≈10.018254161271326637