Farey sequences are sequences of fractions in increasing order in the interval [0;1] whose denominators are less than or equal to a given value. Pick's theorem provides a formula for the calculation of the area of polygonal regions on a lattice with integer coordinates. Although they seem to be quite unrelated topics there is a deep relationship between them. In particular, the fundamental property of Farey sequences can be inferred from Pick's theorem. Conversely, the same property can be used as a starting point of a proof of Pick's theorem. This paper concerns the analysis of such an interesting relationship.