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Barbier's theorem
Barbier's Theorem is a basic result on curves of constant width first proved by Joseph Emile Barbier .
The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. A circle of width (diameter) w has perimeter πw. A Reuleaux triangle of width w consists of three arcs of circles of radius w. Each of these arcs has central angle π/3, so the perimeter of the Reuleaux triangle of width w is equal to ½ the perimeter of a circle of radius w and therefore is equal to πw. A similar analysis of other simple examples such as Reuleaux polygons gives the same answer, and in fact, Barbier's theorem states that the perimeter of any curve of constant width w equals πw.
The analogue of Barbier's theorem for surfaces of constant width is false.
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