The surface of revolution generated by the external catenary between a fixed point 
and its conjugate on the envelope
of the catenary through the fixed point is equal in
area to the surface of revolution generated by its two Lindelof
tangents which cross the axis of rotation at the point
and are calculable from the positions of the points and catenary.
Lindelöf's Catenary Theorem
See also
Catenary, Envelope, Lindelöf's Theorem, Surface of RevolutionExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Lindelöf's Catenary Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LindelofsCatenaryTheorem.html