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 pentagonal rotunda 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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s (). It can be seen as half of an

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 ...

, or as half of a

pentagonal orthobirotunda In geometry, the pentagonal orthobirotunda is one of the Johnson solids (). It can be constructed by joining two pentagonal rotundae () along their decagonal faces, matching like faces. Related polyhedra The pentagonal orthobirotunda is als ...

. It has a total of 17 faces.

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). Th ...

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 ...

circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

, and height are valid 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 aff ...

are

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

, with edge length ''a'': :

V=\left(\frac\left(45+17\sqrt\right)\right)a^3\approx6.91776...a^3

\begin
A&=\left(\frac\sqrt\right)a^2 \\
&=\left(\frac\left(5\sqrt+\sqrt\right)\right)a^2\approx22.3472...a^2
\end

R=\left(\frac\left(1+\sqrt\right)\right)a\approx1.61803...a

H=\left(\sqrt\right)a\approx1.37638...a

Dual polyhedron

The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.

References

External links

* {{Johnson solids Johnson solids