19th Johnson solid
In geometry , the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid .
Construction
The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation .[ 1] equilateral triangles , thirteen squares , and one regular octagon .[ 2] convex polyhedron in which all of the faces are regular polygons is the Johnson solid . The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid
J
19
{\displaystyle J_{19}}
[ 3]
Properties
The surface area of an elongated square cupola
A
{\displaystyle A}
V
{\displaystyle V}
a
{\displaystyle a}
[ 4]
A
=
(
15
+
2
2
+
3
)
a
2
≈ ≈ -->
19.561
a
2
,
V
=
(
3
+
8
2
3
)
a
3
≈ ≈ -->
6.771
a
3
.
{\displaystyle {\begin{aligned}A&=\left(15+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\approx 19.561a^{2},\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\approx 6.771a^{3}.\end{aligned}}}
References
^ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem doi :10.1007/978-93-86279-06-4 , ISBN 978-93-86279-06-4 ^ 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 ^ Francis, Darryl (August 2013), "Johnson solids & their acronyms" , Word Ways , 46 (3): 177 ^ 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
