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 gyrobicupola 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 gyrobicupola In geometry, the pentagonal gyrobicupola is one of the Johnson solids (). Like the pentagonal orthobicupola (), it can be obtained by joining two pentagonal cupolae () along their bases. The difference is that in this solid, the two halves are ...

() 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 cupola In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon. Formulae The fol ...

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

Formulae

The following

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

e 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 are regular, with edge length ''a'': Stephen Wolfram,
Elongated pentagonal gyrobicupola
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 25, 2010. :

V=\frac\left(10+8\sqrt+15\sqrt\right)a^3\approx12.3423...a^3

A=\left(20+\sqrt\right)a^2\approx27.7711...a^2

References

External links

* {{Polyhedron-stub Johnson solids