 Rank3
TypeSegmentotope
SpaceSpherical
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$\frac{\sqrt3}{2} \approx 0.86603$ Dihedral angles5–4: $\arccos\left(\sqrt{\frac{5+\sqrt5}{10}}\right) \approx 31.71747^\circ$ 3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) \approx 20.90516^\circ$ Height$\sqrt{\frac{5-\sqrt5}{10}} \approx 0.52573$ 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:

• $\left(\pm\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0\right),$ • $\left(\pm\frac{\sqrt5+1}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0\right),$ • $\left(\pm\frac{\sqrt5-1}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$ • $\left(\pm\frac12,\,\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$ • $\left(0,\,\sqrt{\frac{5-\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{10}}\right).$ ## 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.