Elongated pentagonal pyramid
| Elongated pentagonal pyramid | |
|---|---|
 | |
| Type | Johnson J8 – J9 – J10 |
| Faces | 5 triangles 5 squares 1 pentagon |
| Edges | 20 |
| Vertices | 11 |
| Vertex configuration | 5(42.5) 5(32.42) 1(35) |
| Symmetry group | C5v, [5], (*55) |
| Rotation group | C5, [5]+, (55) |
| Dual polyhedron | self |
| Properties | convex |
| Net | |
 | |
The elongated pentagonal pyramid is a polyhedron constructed by attaching one pentagonal pyramid onto one of the pentagonal prism's bases, a process known as elongation. It is an example of composite polyhedron.[1][2] This construction involves the removal of one pentagonal face and replacing it with the pyramid. The resulting polyhedron has five equilateral triangles, five squares, and one pentagon as its faces.[3] It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the sixteenth Johnson solid .[4]
For edge length , an elongated pentagonal pyramid has a surface area by summing the area of all faces, and volume by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:[3]
The elongated pentagonal pyramid has a dihedral between its adjacent faces:[5]
- the dihedral angle between square-to-square is the internal angle of its pentagonal base of a prism, 108°;
- the dihedral angle between pentagon-to-square is the right angle, 90°;
- the dihedral angle between triangle-to-triangle is the dihedral angle of a pentagonal pyramid between those, 138.19°; and
- the dihedral angle between triangle-to-square is the sum of the angle between those in a pentagonal pyramid and the angle between the base of and the lateral face of a prism, 127.37°
References
[edit]- ^ 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.
- ^ 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.
- ^ a b 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.
- ^ 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.
- ^ 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.
External links
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