Consider two cylinders as illustrated above (Hubbell 1965) where the cylinders have radii and with , the larger cylinder is oriented along the axis, and where the axes of the two cylinders intersecting at an angle . Then the volume of the region of intersection is given by
(1)
 
(2)
 
(3)
 
(4)
 
(5)

where
(6)

Here, and are complete elliptic integrals of the first and second kinds, respectively, is a hypergeometric function, and is a binomial coefficient.
The intersection of two (or three) right cylinders of equal radii intersecting at right angles is known as the Steinmetz solid.