Elongated triangular pyramid

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Elongated triangular pyramid
Elongated triangular pyramid.png
Type Johnson
J6J7J8
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
Elongated Triangular Pyramid Net.svg
Johnson solid J7. Tetraedro elongado 3D.stl
Johnson solid J7.

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.

Contents

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]

Related Research Articles

<span class="mw-page-title-main">Gyroelongated square bipyramid</span> 17th Johnson solid

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid.

<span class="mw-page-title-main">Gyroelongated square pyramid</span> 10th Johnson solid (13 faces)

In geometry, the gyroelongated square pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in chemistry; for example, the square antiprismatic molecular geometry.

<span class="mw-page-title-main">Square pyramid</span> Pyramid with a square base

In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid, an example of a Johnson solid.

<span class="mw-page-title-main">Triangular cupola</span> Cupola with hexagonal base

In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can be applied to construct many polyhedrons.

<span class="mw-page-title-main">Gyroelongated square bicupola</span> 45th Johnson solid

In geometry, the gyroelongated square bicupola is the Johnson solid constructed by attaching two square cupolae on each base of octagonal antiprism. It has the property of chirality.

<span class="mw-page-title-main">Elongated square pyramid</span> Polyhedron with cube and square pyramid

In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular bipyramid</span> 14th Johnson solid; triangular prism capped with tetrahedra

In geometry, the elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated square bipyramid</span> Cube capped by two square pyramids

In geometry, the elongated square bipyramid is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. It can also be seen as 4 lunes linked together with squares to squares and triangles to triangles. It is also been named the pencil cube or 12-faced pencil cube due to its shape.

<span class="mw-page-title-main">Elongated pentagonal bipyramid</span> 16th Johnson solid; pentagonal prism capped by pyramids

In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Augmented triangular prism</span> 49th Johnson solid

In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.

<span class="mw-page-title-main">Biaugmented triangular prism</span> 50th Johnson solid

In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid. It can be found in stereochemistry in bicapped trigonal prismatic molecular geometry.

<span class="mw-page-title-main">Augmented pentagonal prism</span> 52nd Johnson solid

In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Biaugmented pentagonal prism</span> 53rd Johnson solid

In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.

<span class="mw-page-title-main">Augmented hexagonal prism</span> 54th Johnson solid

In geometry, the augmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism, a metabiaugmented hexagonal prism, or a triaugmented hexagonal prism.

<span class="mw-page-title-main">Elongated triangular cupola</span> Polyhedron with triangular cupola and hexagonal prism

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.

<span class="mw-page-title-main">Elongated triangular orthobicupola</span> Johnson solid with 20 faces

In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

<span class="mw-page-title-main">Elongated triangular gyrobicupola</span> 36th Johnson solid

In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.

<span class="mw-page-title-main">Augmented truncated tetrahedron</span> 65th Johnson solid

In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto an truncated tetrahedron. It is an example of a Johnson solid.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

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. 8489. doi:10.1007/978-93-86279-06-4. ISBN   978-93-86279-06-4.
  2. 1 2 3 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. 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. S2CID   220150682.
  4. Sapiña, R. "Area and volume of the Johnson solid ". Problemas y Ecuaciones (in Spanish). ISSN   2659-9899 . Retrieved 2020-09-09.
  5. 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.