A formula, apparently new, is derived for the spatial rotation of a free rigid body during one periodTof its body angular momentum vector. This formula has the structure of recent phase formulas of Berry [Proc. R. Soc. London Ser. A392, 45–57 (1984); Sci. Am.259(6), 46–55 (1988)] and Hannay [J. Phys. A18, 221–230 (1985)]; namely, it consists of a geometric and a dynamic part. It is derived by applying Stokes’ theorem to a certain closed curve in phase space.