Elongated triangular cupola | |
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Type |
Johnson J17 – J18 – J19 |
Faces | 4
triangles 9 squares 1 hexagon |
Edges | 27 |
Vertices | 15 |
Vertex configuration | 6(42.6) 3(3.4.3.4) 6(3.43) |
Symmetry group | C3v |
Dual polyhedron | - |
Properties | convex |
Net | |
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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.
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. [1] This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon. [2] 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 . [3]
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: [2]
It has the three-dimensional same symmetry as the triangular cupola, the cyclic group of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism: [4]
The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.
Dual elongated triangular cupola | Net of dual |
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The elongated triangular cupola can form a tessellation of space with tetrahedra and square pyramids. [5]
Elongated triangular cupola | |
---|---|
![]() | |
Type |
Johnson J17 – J18 – J19 |
Faces | 4
triangles 9 squares 1 hexagon |
Edges | 27 |
Vertices | 15 |
Vertex configuration | 6(42.6) 3(3.4.3.4) 6(3.43) |
Symmetry group | C3v |
Dual polyhedron | - |
Properties | convex |
Net | |
![]() |
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.
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. [1] This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon. [2] 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 . [3]
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: [2]
It has the three-dimensional same symmetry as the triangular cupola, the cyclic group of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism: [4]
The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.
Dual elongated triangular cupola | Net of dual |
---|---|
![]() |
![]() |
The elongated triangular cupola can form a tessellation of space with tetrahedra and square pyramids. [5]