An -gonal cupola 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 are

(1)

and the coordinates of the top polyhedron vertices are

(2)

where and are the circumradii of the base and top

			(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 (1) and (2) to obtain the coordinates of neighboring bottom and top polyhedron vertices,

			(5)
			(6)

Since all side lengths are ,

(7)

Solving for then gives

(8)

(9)

			(10)
			(11)

Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.

CITE THIS AS:

Weisstein, Eric W. "Cupola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cupola.html