Cupola

An -gonal cupola , where is an integer, is a polyhedron having obliquely oriented triangular and rectangular faces separating an and a regular polygon, each oriented horizontally. The coordinates of the base polyhedron vertices for , 1, ..., are

(1)

and the coordinates of the top polyhedron vertices for , 1, ..., are

(2)

where is the common side length of the base and top, and and are their circumradii,

			(3)
			(4)

and is the height.

A cupola with all unit edge lengths (in which case the triangles become unit equilateral triangles and the rectangles become unit squares) is possible only for , 4, 5, in which case the height can be obtained by letting in the equations (3) and (2) to obtain the coordinates of neighboring bottom and top polyhedron vertices,

			(5)
			(6)

Since all side lengths are ,

(7)

Expanding the squared distance gives

(8)

(9)

Solving for then gives

			(10)
			(11)