Elongated triangular cupola explained

Type:	Johnson
Faces:	4 triangles 9 squares 1 hexagon
Edges:	27
Vertices:	15
Dual:	-
Properties:	convex
Net:	Johnson solid 18 net.png

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

Construction

The elongated triangular cupola is constructed from a hexagonal prism by attaching a triangular cupola onto one of its bases, a process known as the elongation. This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid

J₁₈

Properties

The surface area of an elongated triangular cupola

is the sum of all polygonal face's area. The volume of an elongated triangular cupola can be ascertained by dissecting it into a cupola and a hexagonal prism, after which summing their volume. Given the edge length

, its surface and volume can be formulated as:

\begin A &= \fraca^2 &\approx 13.330a^2, \\ V &= \fraca^3 &\approx 3.777a^3.\end

C_3v

of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism:

the dihedral angle of an elongated triangular cupola between square-to-triangle is that of a triangular cupola between those: 125.3°;
the dihedral angle of an elongated triangular cupola between two adjacent squares is that of a hexagonal prism, the internal angle of its base 120°;
the dihedral angle of a hexagonal prism between square-to-hexagon is 90°, that of a triangular cupola between square-to-hexagon is 54.7°, and that of a triangular cupola between triangle-to-hexagonal is an 70.5°. Therefore, the elongated triangular cupola between square-to-square and triangle-to-square, on the edge where a triangular cupola is attached to a hexagonal prism, is 90° + 54.7° = 144.7° and 90° + 70.5° = 166.5° respectively.

Dual polyhedron

The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.

Related polyhedra and honeycombs

The elongated triangular cupola can form a tessellation of space with tetrahedra and square pyramids.^[1]

Notes and References

Web site: J18 honeycomb.