Rank3
TypeSegmentotope
Notation
Bowers style acronymRapescu
Elements
Faces5 triangles, 5 squares, 1 pentagon
Edges5+5+10
Vertices5+5
Vertex figures5 isosceles trapezoids, edge lengths 1, 2, (1+5)/2, 2
5 butterflies, edge lengths 1, 2, 1, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Dihedral angles5–4: ${\displaystyle \arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }}$
3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Related polytopes
ArmyPentagonal frustum
ConjugatePentagrammic cuploid
Abstract & topological properties
Euler characteristic1
OrientableNo
Genus1
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The retrograde pentagonal cuploid, also called the retrograde pentagonal semicupola or rapescu, is an orbiform polyhedron. It consists of 5 triangles, 5 squares, and 1 pentagon. It is a cuploid based on the pentagon seen as {5/4}, with a pseudo {10/4} base.

## Vertex coordinates

A retrograde pentagonal cuploid of edge length 1 has vertices given by the following coordinates:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}+1}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right).}$

## Related polyhedra

The retrograde pentagonal cuploid can be edge-inscribed into the small ditrigonary icosidodecahedron; it uses triangles and pentagons of the great ditrigonary icosidodecahedron as well as squares of the rhombihedron, the inscribed compound of 5 cubes.