Let \(A\) be a collection of points in \(\mathbb{R}^d\) whose affine span is \(\mathbb{R}^d\), and \(\text{int}(A)\) the set of points in \(A\) that lie in the interior of its convex hull \(\text{conv}(A)\). A subset \(K\subseteq A\) is called free if \(\text{conv} (K)\cap A=K\) and the set of vertices of \(\text{conv}(K)\) is \(K\). The authors prove that \[ \bigl|\text{int}(A) \bigr|= (-1)^{d-1} \sum_{K \text{free}} (-1)^{|K|} |K|, \] a recent conjecture of Ahrens, Gordon, and McMahon, by showing that this formula can be interpreted as a sum of Euler characteristics of certain complexes associated with \(A\), and then computing the homology of these complexes. This method extends to other examples of convex geometries.