The pentagonal rotunda is a
convex polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
with regular polygonal faces. These faces comprise ten
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s, six
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s, and one
regular decagon, making a total of seventeen.
The pentagonal rotunda is an example of
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
, enumerated as the sixth Johnson solid
.
It is another example of a
elementary polyhedron
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids.
Definition and examples
A convex polyhedron is said to be ''composite'' if there exists ...
because by slicing it with a plane, the resulting smaller convex polyhedra do not have regular faces.
The pentagonal rotunda can be regarded as half of an
icosidodecahedron
In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...
, an
Archimedean solid
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
, or as half of a
pentagonal orthobirotunda
In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is an example of Johnson solid.
Construction
The pentagonal orthobirotunda is con ...
, another Johnson solid. Both polyhedrons are constructed by attaching two pentagonal rotundas base-to-base. The difference is one of the pentagonal rotundas is twisted. Other Johnson solids constructed by attaching to the base of a pentagonal rotunda are
elongated pentagonal rotunda
In geometry, the elongated pentagonal rotunda is one of the Johnson solids (''J''21). As the name suggests, it can be constructed by elongating a pentagonal rotunda (''J''6) by attaching a decagonal prism to its base. It can also be seen as an el ...
,
gyroelongated pentagonal rotunda
In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (''J''25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (''J''6) by attaching a decagonal antiprism to its base. It can also be s ...
,
pentagonal orthocupolarotunda
In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by joining a pentagonal cupola () and a pentagonal rotunda () along their decagonal bases, matching the pentagonal face ...
,
pentagonal gyrocupolarotunda
In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (). Like the pentagonal orthocupolarotunda (), it can be constructed by joining a pentagonal cupola () and a pentagonal rotunda () along their decagonal bases. The differ ...
,
elongated pentagonal orthocupolarotunda
In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda () by inserting a decagonal prism between its halves. Rotating eit ...
,
elongated pentagonal gyrocupolarotunda
In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda () by inserting a decagonal prism between its halves. Rotating eith ...
,
elongated pentagonal orthobirotunda
In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (). Its Conway polyhedron notation iat5jP5 As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda () by inserting a decagonal prism ...
,
elongated pentagonal gyrobirotunda
In geometry, the elongated pentagonal gyrobirotunda or elongated icosidodecahedron is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimede ...
,
gyroelongated pentagonal cupolarotunda
In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda ( or ) by inserting a decagonal antiprism between its two halves.
...
, and
gyroelongated pentagonal birotunda
In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either or the icosidodecahedron) by inserting a decagonal antiprism betwee ...
.
As an above, the surface area
and volume
of a pentagonal rotunda are the following:
References
External links
*
{{Johnson solids
Johnson solids