Elongated pentagonal cupola

Elongated pentagonal cupola
Type	Johnson J19 – J20 – J21
Faces	5 triangles 15 squares 1 pentagon 1 decagon
Edges	45
Vertices	25
Vertex configuration	10(42.10) 10(3.43) 5(3.4.5.4)
Symmetry group	C5v
Properties	convex, composite
Net

The elongated pentagonal cupola is a polyhedron, constructed by attaching pentagonal cupola to a decagonal prism to its base. It is a Johnson solid

Construction

The elongated pentagonal cupola is constructed from a ten-sided prism by attaching a pentagonal cupola onto one of its bases, a process known as the elongation. This cupola covers the decagon, so that the resulting polyhedron has five equilateral triangles, fifteen squares, one regular pentagon, and one regular decagon. ^[1] By such a construction, the elongated pentagonal cupola is composite. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated pentagonal cupola is one of them, enumerated as the twentieth Johnson solid ${\displaystyle J_{20}}$. ^[2]

Properties

3D model of an elongated pentagonal cupola

The surface area of an elongated square cupola ${\displaystyle A}$ is the sum of the area of all faces: five equilateral triangles, fifteen squares, one regular pentagon, and one regular decagon. Its volume ${\displaystyle V}$ can be ascertained by dissecting it into both a pentagonal cupola and a regular decagon, and then adding their volume. Let ${\displaystyle a}$ be the edge length of an elongated pentagonal cupola, its surface area and volume are: ^[3] $${\displaystyle {\begin{aligned}A&=\left({\frac {1}{4}}\left(60+{\sqrt {10\left(80+31{\sqrt {5}}+{\sqrt {2175+930{\sqrt {5}}}}\right)}}\right)\right)a^{2}\\&\approx 26.5797a^{2},\\V&=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)\right)a^{3}\\&\approx 10.0183a^{3}.\end{aligned}}}$$

References

1. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR   0290245 .
2. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
3. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics , 18: 169–200, doi: 10.4153/cjm-1966-021-8 , MR   0185507, S2CID   122006114, Zbl   0132.14603 .

External links

• Weisstein, Eric W., "Elongated pentagonal cupola" ("Johnson solid") at MathWorld .