The basic question of geometric combinatorics is to determine whether a sufficiently large set determines many classes of geometric objects, defined by point tuples. In Euclidean plane, such geometric objects are, for instance, line segments or triangles. Analogues of these questions in vector space over finite fields have also been studied. In the talk, I will discuss the distribution of areas of triangles determined by subsets of the finite plane. The finite field version of Beck’s theorem will also be developed, which will be combined with Fourier analytic techniques to prove the main result about areas of triangles.