In
geometry, the elongated square cupola 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
square cupola () by attaching an
octagonal prism to its base. The solid can be seen as a
rhombicuboctahedron with its "lid" (another
square cupola) removed.
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,
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 ...
and
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 ...
can be used if all
faces are
regular, with edge length ''a'':
:
:
:
Dual polyhedron
The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.
Related polyhedra and honeycombs
The elongated square cupola forms space-filling
honeycombs with
tetrahedra and
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
s; with cubes and
cuboctahedra
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...
; and with tetrahedra,
elongated square pyramid
In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically ( ...
s, and
elongated square bipyramid
In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.
It has been named ...
s. (The latter two units can be decomposed into cubes and
square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...
s.)
References
External links
*
{{Polyhedron-stub
Johnson solids