Elongated cupola

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Set of elongated cupolae
Elongated pentagonal cupola.png
Example pentagonal form
Faces n triangles
3n squares
1 n-gon
1 2n-gon
Edges 9n
Vertices 5n
Symmetry group Cnv, [n], (*nn)
Rotational group Cn, [n]+, (nn)
Dual polyhedron
Properties convex

In geometry, the elongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an n-gonal prism.

There are three elongated cupolae that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. Adjoining a triangular prism to a cube also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. Higher forms can be constructed without regular faces.


name faces
elongated triangular prism 2 triangles, 6+1 squares
Elongated triangular cupola.png elongated triangular cupola (J18) 3+1 triangles, 9 squares, 1 hexagon
Elongated square cupola.png elongated square cupola (J19) 4 triangles, 12+1 squares, 1 octagon
Elongated pentagonal cupola.png elongated pentagonal cupola (J20) 5 triangles, 15 squares, 1 pentagon, 1 decagon
elongated hexagonal cupola 6 triangles, 18 squares, 1 hexagon, 1 dodecagon

See also[edit]


  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.