Prisms & Antiprisms

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(Uniform #77) Pentagonal Antiprism
Vertices:  10  (10[4])
Faces:12  (10 equilateral triangles + 2 regular pentagons)
Edges:20
Symmetry:  5-fold Antiprismatic  (D5v)
Pentagon-Triangle Angle:  acos(−sqrt(15*(5−2*sqrt(5)))/15)    ≈100.812316964 degrees
Triangle-Triangle Angle:  acos(−sqrt(5)/3)    ≈138.189685104 degrees
Dual Solid:  Pentagonal Trapezohedron
(values below based on edge length = 1)
Circumscribed Radius:  sqrt(2*(5+sqrt(5)))/4    ≈0.95105651629515357212
Midscribed Radius:  (1+sqrt(5))/4    ≈0.80901699437494742410
Pentagon Center Radius:  sqrt(10*(5+sqrt(5)))/20    ≈0.42532540417601996609
Triangle Center Radius:  (3*sqrt(3)+sqrt(15))/12    ≈0.75576131407617073048
Volume:(5+2*sqrt(5))/6    ≈1.5786893258332632321


References:[1]Johannes Kepler, Harmonices Mundi (1619).
[2]Johannes Kepler with E. J. Aiton, A. M. Duncan, and J. V. Field, translators,
The Harmony of the World, American Philosophical Society (1997).