Elongated square pyramid

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Elongated square pyramid
Elongated square pyramid.png
Type Johnson
J7J8J9
Faces 4 triangles
1+4 squares
Edges 16
Vertices 9
Vertex configuration

Symmetry group
Dihedral angle (degrees)
  • triangle-to-triangle: 109.47°
  • square-to-square: 90°
  • triangle-to-square: 144.74°
Dual polyhedron self-dual [1]
Properties convex, composite
Net
Elongated Square Pyramid Net.svg

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.

Contents

Construction

The elongated square pyramid is a composite, since it can constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid. [2] [3] One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite. [4]

A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as , the fifteenth Johnson solid. [5]

Properties

Given that is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is . Therefore, the height of an elongated square bipyramid is: [6] Its surface area can be calculated by adding all the area of four equilateral triangles and four squares: [4] Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them: [4]

3D model of a elongated square pyramid. Piramide cuadrada elongada.stl
3D model of a elongated square pyramid.

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube: [7]

See also

References

  1. Draghicescu, Mircea. "Dual Models: One Shape to Make Them All". In Torrence, Eva; Torrence, Bruce; Séquin, Carlo H.; McKenna, Douglas; Fenyvesi, Kristóf; Sarhangi, Reza (eds.). Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture (PDF). pp. 635–640.
  2. Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
  3. 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.
  4. 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.
  5. 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.
  6. Sapiña, R. "Area and volume of the Johnson solid ". Problemas y Ecuaciones (in Spanish). ISSN   2659-9899 . Retrieved 2020-09-09.
  7. 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.