The rapid and accurate assessment of building damage states using only post-event remote sensing data is critical when performing loss estimation in earthquake emergency response. Damaged roof detection is one of the most efficient methods of assessing building damage. In particular, airborne LiDAR is often used to detect roofs damaged by earthquakes, especially for certain damage types, due to its ability to rapidly acquire accurate 3D information on individual roofs. Earthquake-induced roof damages are categorized into surface damages and structural damages based on the geometry features of the debris and the roof structure. However, recent studies have mainly focused on surface damage; little research has been conducted on structural damage. This paper presents an original 3D shape descriptor of individual roofs for detecting roofs with surface damage and roofs exhibiting structural damage by identifying spatial patterns of compact and regular contours for intact roofs, as well as jagged and irregular contours for damaged roofs. The 3D shape descriptor is extracted from building contours derived from airborne LiDAR point clouds. First, contour clusters are extracted from contours that are generated from a dense DSM of individual buildings derived from point clouds. Second, the shape chaos indexes of contour clusters are computed as the information entropy through a contour shape similarity measurement between two contours in a contour cluster. Finally, the 3D shape descriptor is calculated as the weighted sum of the shape chaos index of each contour cluster corresponding to an individual roof. Damaged roofs are detected solely using the 3D shape descriptor with the maximum entropy threshold. Experiments using post-event airborne LiDAR point clouds of the 2010 Haiti earthquake suggest that the proposed damaged roof detection technique using the proposed 3D shape descriptor can detect both roofs exhibiting surface damage and roofs exhibiting structural damage with a high accuracy.