Our research focuses on establishing a mathematical basis for examining whether a district has likely been gerrymandered. As of right now, a judicial precedent exists that one cannot define an exact notion of district “compactness” for determining gerrymandering, but you “know it when you see it”. Academic papers have been released attempting to prove a geometric basis for the “know it when you see it” phenomenon centering around a variety of distinct compactness metrics for shapes. Our paper explores the effectiveness of a medial axis-based compactness measure and the extended distance function that can be calculated based on the medial axis. We make the case that this measure both objectively performs well in identifying gerrymandered districts and also holds a more accurate representation of how humans see shapes than previously published approaches to computing compactness. Through our research, we compare the medial axis-based compactness measure to other published methods of assessing compactness, and we analyze the performance of our model compared to others in analyzing states that have been extensively covered in the media as regions with gerrymandered districts.