geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the elongated pentagonal gyrobirotunda is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...

(one of the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s), by inserting a

decagonal prism In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, a ...

between its congruent halves. Rotating one of the

pentagonal rotunda In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. Formulae The following formulae for volume, surface ar ...

e () through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda ().

Formulae

The following formulae for

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

and

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

can be used if all

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

are regular, with edge length ''a'':

Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...

,
Elongated pentagonal gyrobirotunda
from

Wolfram Alpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mathe ...

. Retrieved July 26, 2010. :

V=\frac\left(45+17\sqrt+15\sqrt\right)a^3 \approx 21.5297 a^3

A=\left(10+\sqrt\right)a^2 \approx 39.306 a^2

References

External links

* {{Polyhedron-stub Johnson solids