Archimedes' hat-box theorem equates the lateral surface area of a spherical segment with that cut
from its circumscribing cylinder. Enclose a sphere
in a cylinder and cut out a spherical
segment by slicing twice perpendicularly to
the cylinder's axis. Then the lateral surface
area of the spherical segment 
is equal to the lateral surface
area cut out of the cylinder 
by the same slicing planes, i.e.,
 |
where 
is the radius of the cylinder
(and tangent sphere) and 
is the height of the cylindrical
(and spherical) segment.
Ghomi et al. (2026, p. 1) proved a converse using only one fixed band width. Let 
be a connected smooth closed
surface embedded in 
, and fix 
smaller than the minimum distance between parallel supporting hyperplanes of 
. If every band cut from 
by parallel planes
separated by 
and lying between these supporting hyperplanes
has surface area 
, then 
is a translate of the unit sphere.
