- Source: Pentagonal gyrobicupola
The pentagonal gyrobicupola is a polyhedron that is constructed by attaching two pentagonal cupolas base-to-base, each of its cupolas is twisted at 36°. It is an example of a Johnson solid and a composite polyhedron.
Construction
The pentagonal gyrobicupola is a composite polyhedron: it is constructed by attaching two pentagonal cupolas base-to-base. This construction is similar to the pentagonal orthobicupola; the difference is that one of cupolas in the pentagonal gyrobicupola is twisted at 36°, as suggested by the prefix gyro-. The resulting polyhedron has the same faces as the pentagonal orthobicupola does: those cupolas cover their decagonal bases, replacing it with eight equilateral triangles, eight squares, and two regular pentagons. A convex polyhedron in which all of its faces are regular polygons is the Johnson solid. The pentagonal gyrobicupola has such these, enumerating it as the thirty-first Johnson solid
J
31
{\displaystyle J_{31}}
.
Properties
Because it has a similar construction as the pentagonal orthobicupola, the surface area of a pentagonal gyrobicupola
A
{\displaystyle A}
is the sum of polygonal faces' area, and its volume
V
{\displaystyle V}
is twice the volume of a pentagonal cupola for which slicing it into those:
A
=
20
+
100
+
10
5
+
10
75
+
30
5
2
a
2
≈
17.771
a
2
,
V
=
5
+
4
5
3
a
3
≈
4.648
a
3
.
{\displaystyle {\begin{aligned}A&={\frac {20+{\sqrt {100+10{\sqrt {5}}+10{\sqrt {75+30{\sqrt {5}}}}}}}{2}}a^{2}\approx 17.771a^{2},\\V&={\frac {5+4{\sqrt {5}}}{3}}a^{3}\approx 4.648a^{3}.\end{aligned}}}
References
External links
Weisstein, Eric W., "Pentagonal gyrobicupola" ("Johnson solid") at MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Pentagonal gyrobicupola
- Elongated pentagonal gyrobicupola
- Elongated pentagonal orthobicupola
- Pentagonal orthobicupola
- Johnson solid
- Pentagonal cupola
- List of polygons, polyhedra and polytopes
- List of mathematical shapes
- Bicupola
- J39
