Cupola (geometry)
Encyclopedia
Pentagonal cupola (example)
Type Set of cupolas
Faces n triangles,
n square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s
1 n-agon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

,
1 2n-agon
Edges 5n
Vertices 3n
Symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

Cnv, [1,n], (*nn)
Dual polyhedron
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

?
Properties convex

In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cupola is a solid formed by joining two polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

s, one (the base) with twice as many edges as the other, by an alternating band of triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s and rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

s. If the triangles are equilateral and the rectangles are square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s, while the base and its opposite face are regular polygons, the triangular
Triangular cupola
In geometry, the triangular cupola is one of the Johnson solids . It can be seen as half a cuboctahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966.-Formulae:...

, square
Square cupola
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids . It can be obtained as a slice of the rhombicuboctahedron...

, and pentagonal
Pentagonal cupola
In geometry, the pentagonal cupola is one of the Johnson solids . It can be obtained as a slice of the rhombicosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....

cupolae all count among the Johnson solid
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s, and can be formed by taking sections of the cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

, rhombicuboctahedron
Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

, and rhombicosidodecahedron
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....

, respectively.

A cupola can be seen as a prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

where one of the polygons has been collapsed in half by merging alternate vertices.

Cupolae are a subclass of the prismatoid
Prismatoid
In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes....

s.

## Examples

The above-mentioned three polyhedra are the only non-trivial cupolae with regular faces: The "hexagonal cupola" is a plane figure, and the triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....

might be considered a "cupola" of degree 2 (the cupola of a line segment and a square). However, cupolae of higher-degree polygons may be constructed with irregular triangular and rectangular faces.

## Coordinates of the vertices

The definition of the cupola does not require the base (or the side opposite the base, which can be called the top) to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry, Cnv. In that case, the top is a regular n-gon, while the base is either a regular 2n-gon or a 2n-gon which has two different side lengths alternating and the same angles as a regular 2n-gon. It is convenient to fix the coordinate system so that the base lies in the xy-plane, with the top in a plane parallel to the xy-plane. The z-axis is the n-fold axis, and the mirror planes pass through the z-axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. (If n is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if n is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated V1 through V2n, while the vertices of the top polygon can be designated V2n+1 through V3n. With these conventions, the coordinates of the vertices can be written as:
• V2j-1: (rbcos [2π(j-1)/n + α], rbsin [2π(j-1)/n + α], 0)
• V2j: (rbcos (2πj/n - α), rbsin (2πj/n - α), 0)
• V2n+j: (rtcos (πj/n), rtsin (πj/n), h)

where j=1, 2, …, n.

Since the polygons V1V2V2n+2V2n+1, etc. are rectangles, this puts a constraint on the values of rb, rt, and α. The distance V1V2 is equal to

rb{[cos (2π/n - α) – cos α]2 + [sin (2π/n - α) - sin α] 2}1/2
rb{[cos 2 (2π/n - α) – 2cos (2π/n - α)cos α + cos2 α] + [sin 2 (2π/n - α) – 2 sin (2π/n - α)sin α + sin 2α]}1/2
rb{2[1 – cos (2π/n - 2α)]}1/2,

while the distance V2n+1V2n+2 is equal to

rt{[cos (π/n) – 1]2 + sin2(π/n)}1/2
rt{2[1 – cos (π/n)]}1/2.

These are to be equal, and if this common edge is denoted by s,

rb = s/{2[1 – cos (2π/n - 2α)]}1/2

rt= s/{2[1 – cos (π/n)]}1/2

These values are to be inserted into the expressions for the coordinates of the vertices given earlier.

## Hypercupolas

The hypercupolas are a family of convex nonuniform polychora (here four-dimensional figures), analogous to the cupolas. Each one's bases are a Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

and its expansion
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

.