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-dual [1]
Properties	convex
Net

3D model of an elongated pentagonal pyramid

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. ^{[2] [3]} 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. ^[4] It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the ninth Johnson solid ${\displaystyle J_{9}}$. ^[5]

For edge length ${\displaystyle \ell }$, an elongated pentagonal pyramid has a surface area ${\displaystyle A}$ by summing the area of all faces, and volume ${\displaystyle V}$ by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism: ^[4] $${\displaystyle {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\ell ^{2}\approx 8.886\ell ^{2},\\V&={\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\ell ^{3}\approx 2.022\ell ^{3}.\end{aligned}}}$$

The elongated pentagonal pyramid has a dihedral between its adjacent faces: ^[6]

• the dihedral angle between adjacent squares is the internal angle of the prism's pentagonal base, 108°;
• the dihedral angle between the pentagon and a square is the right angle, 90°;
• the dihedral angle between adjacent triangles is that of a regular icosahedron, 138.19°; and
• the dihedral angle between a triangle and an adjacent 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

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. 84–89. doi:10.1007/978-93-86279-06-4. ISBN   978-93-86279-06-4.
4. 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. 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

• Weisstein, Eric W., "Johnson solid" ("Elongated pentagonal pyramid") at MathWorld .