Elongated triangular pyramid
Polyhedron constructed with tetrahedra and a triangular prism From Wikipedia, the free encyclopedia
In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.
Elongated triangular pyramid | |
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Type | Johnson J6 – J7 – J8 |
Faces | 4 triangles 3 squares |
Edges | 12 |
Vertices | 7 |
Vertex configuration | 1(33) 3(3.42) 3(32.42) |
Symmetry group | C3v, [3], (*33) |
Rotation group | C3, [3]+, (33) |
Dual polyhedron | self |
Properties | convex |
Net | |
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Construction
The elongated triangular pyramid is constructed from a triangular prism by attaching regular tetrahedron onto one of its bases, a process known as elongation.[1] The tetrahedron covers an equilateral triangle, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three squares as its faces.[2] A convex polyhedron in which all of the faces are regular polygons is called the Johnson solid, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid .[3]
Properties
An elongated triangular pyramid with edge length has a height, by adding the height of a regular tetrahedron and a triangular prism:[4] Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:[2] and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:[2]:
It has the three-dimensional symmetry group, the cyclic group of order 6. Its dihedral angle can be calculated by adding the angle of the tetrahedron and the triangular prism:[5]
- the dihedral angle of a tetrahedron between two adjacent triangular faces is ;
- the dihedral angle of the triangular prism between the square to its bases is , and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is ;
- the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle .
References
External links
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