His principal contributions were on irrational lengths, which was included in Book X of Euclid's Elements and proving that there are precisely five regular convex polyhedra.
Cyrene was a prosperous Greek colony on the coast of North Africa, in what is now Libya, on the eastern end of the Gulf of Sidra.
Theodorus had explored the theory of incommensurable quantities, and Theaetetus continued those studies with great enthusiasm; specifically, he classified various forms of irrational numbers according to the way they are expressed as square roots.
p. 18, footnote 67: I find it essential to note that the historians of mathematics who take for granted that Theaitetos was still active in the 370s must be wrong.
When Plato wrote his first draft of the dialogue, presumably in the 370s, he saw this friend of his youth from an idealizing perspective, somewhat in the way he remembered his uncle Charmides.