With upcoming breakthroughs in free-form display technologies, new user interface design challenges have emerged. Here, we investigate a question, which has been widely explored on traditional GUIs but unexplored on non-rectangular interfaces: what are the user strategies in terms of visual search when information is not presented in a traditional rectangular layout? To achieve this, we present two complementary studies investigating eye movements in different visual search tasks. Our results unveil which areas are seen first according to different visual structures. By doing so we address the question of where to place relevant content for the UI designers of non-rectangular displays.