Barbier's theorem

[2] The most familiar examples of curves of constant width are the circle and the Reuleaux triangle.

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 half 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.

If K is a body of constant width w, then the Minkowski sum of K and its 180° rotation is a disk with radius w and perimeter 2πw.

[3] Alternatively, the theorem follows immediately from the Crofton formula in integral geometry according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings.

Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length.

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

[5] Instead, Barbier's theorem generalizes to bodies of constant brightness, three-dimensional convex sets for which every two-dimensional projection has the same area.