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Elongated square cupola

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19th Johnson solid
Elongated square cupola
TypeJohnson
J18J19J20
Faces4 triangles
13 squares
1 octagon
Edges36
Vertices20
Vertex configuration8(4.8)
4+8(3.4)
Symmetry groupC4v
Dual polyhedron-
Propertiesconvex
Net

In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.

Construction

The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation. This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid J 19 {\displaystyle J_{19}} .

Properties

The surface area of an elongated square cupola A {\displaystyle A} is the sum of all polygonal faces' area. Its volume V {\displaystyle V} can be ascertained by dissecting it into both square cupola and regular octagon, and then adding their volume. Given the elongated triangular cupola with edge length a {\displaystyle a} , its surface area and volume are: A = ( 15 + 2 2 + 3 ) a 2 19.561 a 2 , V = ( 3 + 8 2 3 ) a 3 6.771 a 3 . {\displaystyle {\begin{aligned}A&=\left(15+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\approx 19.561a^{2},\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\approx 6.771a^{3}.\end{aligned}}}

References

  1. Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, p. 84–89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.
  2. 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.
  3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
  4. 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

Johnson solids
Pyramids, cupolae and rotundae
Modified pyramids
Modified cupolae and rotundae
Augmented prisms
Modified Platonic solids
Modified Archimedean solids
Other elementary solids
(See also List of Johnson solids, a sortable table)
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