He then spent virtually his entire career as a member of the Brown University Mathematics Department, where he eventually retired with the title of Professor Emeritus.
In 1987, he and his Brown colleague Wendell Fleming won the American Mathematical Society's Steele Prize "for their pioneering work in Normal and Integral currents.
A particularly noteworthy early accomplishment (improving earlier work of Abram Besicovitch) was the characterization of purely unrectifiable sets as those which "vanish" under almost all projections.
[FZ73] In his first published paper, written with his Ph.D. advisor Anthony Morse, Federer proved the Federer–Morse theorem which states that any continuous surjection between compact metric spaces can be restricted to a Borel subset so as to become an injection, without changing the image.
The Steiner formula formed a fundamental precedent for Federer's work; it established that the volume of a neighborhood of a convex set in Euclidean space is given by a polynomial.
He proved the Steiner formula for this class, identifying generalized quermassintegrals (called curvature measures by Federer) as the coefficients.
[FF60] In their work, they showed that Plateau's problem for minimal surfaces can be solved in the class of integral currents, which may be viewed as generalized submanifolds.
[F75] A particular result detailed in Federer's book is that area-minimizing minimal hypersurfaces of Euclidean space are smooth in low dimensions.