Elongated triangular pyramid
| Elongated triangular pyramid | |
|---|---|
 | |
| 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-dual[1] |
| Properties | convex |
| Net | |
 | |
File:Tetraedro elongado 3D.stl
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.
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.[2] 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.[3] 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 .[4]
Properties
An elongated triangular pyramid with edge length has a height, by adding the height of a regular tetrahedron and a triangular prism:[5] Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:[3] and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:[3]:
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:[6]
- 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
- ↑ Draghicescu, Mircea. "Dual Models: One Shape to Make Them All". Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture. pp. 635–640. https://archive.bridgesmathart.org/2016/bridges2016.
- ↑ 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. https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84.
- ↑ 3.0 3.1 3.2 Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8.
- ↑ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62.
- ↑ Sapiña, R.. "Area and volume of the Johnson solid " (in es). Problemas y Ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html. Retrieved 2020-09-09.
- ↑ "Convex polyhedra with regular faces". Canadian Journal of Mathematics 18: 169–200. 1966. doi:10.4153/cjm-1966-021-8.
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