Barbier's theorem states that every convex curve of constant width 
has perimeter
 |
Thus every curve of constant width has the same perimeter as a circle
of diameter 
, even though such a curve need not be a circle.
To see the result, let 
denote the distance between the two parallel supporting
lines perpendicular to
direction 
.
Cauchy's perimeter formula gives 
. For a curve
of constant width, 
,
and the stated formula follows immediately.
