Pentagonal gyrocupolarotunda

{{Infobox polytope The pentagonal gyrocupolarotunda, or pegycuro, is one of the 92 Johnson solids (J33). It consists of 5+5+5 triangles, 5 squares, and 1+1+5 pentagons. It can be constructed by attaching a pentagonal cupola and a pentagonal rotunda at their decagonal bases, such that the two pentagonal bases are rotated 36º with respect to each other.
 * type=CRF
 * img=Pentagonal gyrocupolarotunda 2.png
 * 3d=J33 pentagonal gyrocupolarotunda.stl
 * dim = 3
 * obsa = Pegycuro
 * faces = 5+5+5 triangles, 5 squares, 1+1+5 pentagons
 * edges = 5+5+5+5+10+10+10
 * vertices = 5+5+5+10
 * verf = 5 trapezoids, edge lengths 1, $\sqrt{2}$, (1+$\sqrt{5}$}0/2, $\sqrt{2}$
 * verf2 = 10 rectangles, edge lengths 1 and (1+)/2
 * verf3 = 10 irregular tetragons, edge lengths 1, 1, $\sqrt{2}$, (1+$\sqrt{5}$)/2
 * coxeter = xoxo5ofxx&#xt
 * army=Pegycuro
 * reg=Pegycuro
 * symmetry = H2×I, order 10
 * volume = 5(11+5$\sqrt{5}$)/12 ≈ 9.24181
 * dih = 3–4: acos(–($\sqrt{3}$+$\sqrt{15}$)/6) ≈ 159.09484º
 * dih2 = 4–5 cupolaic: acos(–$\sqrt{(5+√5)/10}$) ≈ 148.28253º
 * dih3 = 3–5: acos(–($\sqrt{5+2√5)/15}$) ≈ 142.62263º
 * dih4 = 3–3: acos(–$\sqrt{5}$/5) ≈ 116.56505º
 * dih5 = 4–5 join: acos(–$\sqrt{(25–11√5)/50}$) ≈ 95.15242º
 * smm = Yes
 * dual = Pentadeltodecatrapezopentadeltopentarhombic icosipentahedron
 * conjugate = Retrograde pentagrammic gyrocupolarotunda
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

If the cupola and rotunda are joined such that the bases are in the same orientation, the result is the pentagonal orthocupolarotunda.

Vertex coordinates
A pentagonal gyrocupolarotunda of edge length 1 has vertices given by the following coordinates:


 * (0, $\sqrt{(5+√5)/10}$, $\sqrt{(5+2√5)/5}$),
 * (±1/2, —$\sqrt{(5+2√5)/20}$, $\sqrt{(5+2√5)/5}$),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{(5–√5)/40}$, $\sqrt{(5+2√5)/5}$),
 * (0, –$\sqrt{(5+2√5)/5}$, $\sqrt{(5+√5)/10}$),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{(25+11√5)/40}$, $\sqrt{(5+√5)/10}$),
 * (±(3+$\sqrt{5}$)/4, –$\sqrt{(5+√5)/40}$, $\sqrt{(5+√5)/10}$),
 * (±1/2, ±$\sqrt{(5+2√5)/4}$, 0),
 * (±(3+$\sqrt{5}$)/4, ±$\sqrt{(5+√5)/8}$, 0),
 * (±(1+$\sqrt{5}$)/2, 0, 0).
 * (±1/2, $\sqrt{(5+2√5)/20}$, –$\sqrt{(5–√5)/10}$),
 * (±(1+$\sqrt{5}$)/4, –$\sqrt{(5+√5)/40}$, –$\sqrt{(5–√5)/10}$),
 * (0, –$\sqrt{(5+√5)/10}$, –$\sqrt{(5–√5)/10}$),

Related polyhedra
A decagonal prism can be inserted between the two halves of the pentagonal gyrocupolarotunda to produce the elongated pentagonal gyrocupolarotunda.