Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see that the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. The talk will be based on joint works with Bossinger, Magee, Najera-Chavez.