He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture).

The family moved to Budapest, when Fejes Tóth was five; there he attended elementary school and high school—the Széchenyi István Reálgimnázium—where his interest in mathematics began.

As a freshman, he developed a generalized solution regarding Cauchy exponential series, which he published in the proceedings of the French Academy of Sciences—1935.

[8] His distinguished academic career allowed him to travel abroad beyond the Iron Curtain to attend international conferences and teach at various universities, including those at Freiburg; Madison, Wisconsin; Ohio; and Salzburg.

[11] Imre Bárány credited Fejes Tóth with several influential proofs in the field of discrete and convex geometry, pertaining to packings and coverings by circles, to convex sets in a plane and to packings and coverings in higher dimensions, including the first correct proof of Thue's theorem.

He credits Fejes Tóth, along with Paul Erdős, as having helped to "create the school of Hungarian discrete geometry.

He emphasized that, at the time of this work, the problem of the upper bound for the density of a packing of equal spheres was still unsolved.

The approach that Fejes Tóth suggested in that work, which translates as "packing [of objects] in a plane, on a sphere and in a space", provided Thomas Hales a basis for a proof of the Kepler conjecture in 1998.

The Kepler conjecture, named after the 17th-century German mathematician and astronomer Johannes Kepler, says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.

[25] In 2015, the year of Fejes Tóth's centennial birth anniversary, the prize was awarded to Károly Bezdek of the University of Calgary in a ceremony held on 19 June 2015 in Veszprém, Hungary.