The purpose of this talk is to demonstrate how certain problems in number theory can be solved via geometry. In particular, we will focus on the Farey sequence of order n, which is the increasing sequence of fractions whose denominators in lowest terms are all at most n. We will observe some remarkable features of these sequences for various n, and rephrase our observations and conjectures as problems in geometry. This will motivate us to introduce and prove Pick´s theorem, which provides a simple formula for computing the area of various polygons in the plane whose vertices have integer coordinates.