Barbier's theorem

Every plane convex curve of constant width has perimeter equal to π times its width, regardless of its shape.

Cauchy's perimeter formula: A formula expressing a plane convex body's perimeter as the integral of its projection widths over directions. It turns the constant projection width into the perimeter value.
Circle: The plane curve consisting of all points at a fixed distance from a center. Its circumference is 2πr, exactly π times its width 2r.
Joseph-Émile Barbier: A French mathematician known for work in geometry, probability, and mathematical physics. The theorem bears his name and appeared in his work on constant-width curves.
Wankel engine: A rotary internal-combustion engine whose triangular rotor turns within a shaped housing. Its rotor uses a rounded triangular form related to constant-width geometry, though not itself a constant-width curve.
Isoperimetric inequality: The inequality stating that among plane regions of fixed perimeter, the circle encloses the greatest area. Unlike Barbier's equality for constant width, it compares area and perimeter across all plane regions.
Support function: A function giving the signed distance from a chosen origin to a convex body's supporting line in each direction. The difference of opposite support values encodes each directional width.
Reuleaux triangle: A constant-width shape formed from three circular arcs centered at the vertices of an equilateral triangle. It is a nonsmooth example with the same perimeter as a circle of equal width.
Constant-width curve: A plane convex body whose distance between parallel supporting lines is the same in every direction. The study of these curves supplied the problem that the theorem resolves.
Coin: A small, flat piece of metal or other material issued as money. Constant-width coin outlines can roll smoothly while retaining a noncircular shape.
Constant-width body: A convex body in Euclidean space whose distance between parallel supporting hyperplanes is direction-independent. This higher-dimensional generalization does not inherit the planar perimeter statement unchanged.

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Blaschke–Lebesgue theoremRelated: It shows that the theorem’s competing bodies already share the same perimeter.