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 π''w''. 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 π''w''. 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 π''w''.
The analogue of Barbier's theorem for surfaces of constant width is false.

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External links

★ The Theorem of Barbier (Java) at cut-the-knot