Augmented pentagonal prism

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.

Augmented pentagonal prism
Type	Johnson J51 – J52 – J53
Faces	4 triangles 4 squares 2 pentagons
Edges	19
Vertices	11
Vertex configuration	2+4(42.5) 1(34) 4(32.4.5)
Symmetry group	C2v
Properties	convex
Net

Construction

The augmented pentagonal prism can be constructed from a pentagonal prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation.^[1] This square pyramid covers the square face of the prism, so the resulting polyhedron has four equilateral triangles, four squares, and two regular pentagons as its faces.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the augmented pentagonal prism is among them, enumerated as 52nd Johnson solid ${\displaystyle J_{52}}$.^[3]

Properties

An augmented pentagonal prism with edge length ${\displaystyle a}$ has a surface area, calculated by adding the area of four equilateral triangles, four squares, and two regular pentagons:^[2]$${\displaystyle {\frac {8+2{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}}{2}}a^{2}\approx 9.173a^{2}.}$$ Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:^[2]$${\displaystyle {\frac {\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}{12}}a^{3}\approx 1.9562a^{3}.}$$

The dihedral angle of an augmented pentagonal prism can be calculated by adding the dihedral angle of an equilateral square pyramid and the regular pentagonal prism:^[4]

• the dihedral angle of an augmented pentagonal prism between two adjacent triangular faces is that of an equilateral square pyramid between two adjacent triangular faces, ${\textstyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.5^{\circ }}$,
• the dihedral angle of an augmented pentagonal prism between two adjacent square faces is the internal angle of a regular pentagon ${\textstyle {\frac {3\pi }{5}}=108^{\circ }}$.
• the dihedral angle of an augmented pentagonal prism between square-to-pentagon is that of a regular pentagonal prism between its base and its lateral faces ${\textstyle {\frac {\pi }{2}}=90^{\circ }}$.
• the dihedral angle of an augmented pentagonal prism between pentagon-to-triangle is ${\textstyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.7^{\circ }}$, for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face ${\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.7^{\circ }}$, and the dihedral angle of a regular pentagonal prism between its base and its lateral face.
• the dihedral angle of an augmented pentagonal prism between square-to-triangle is ${\textstyle \arctan \left({\sqrt {2}}\right)+{\frac {3\pi }{5}}\approx 162.7^{\circ }}$, for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face, and the dihedral angle of a regular pentagonal prism between two adjacent squares.

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. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
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. MR 0290245.
3. Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
4. 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". MathWorld.
  • Weisstein, Eric W. "Augmented pentagonal prism". MathWorld.