As a common and critical medical image analysis task, deep learning based biomedical image segmentation is hindered by the dependence on costly fine-grained annotations. To alleviate this data dependence, in this paper, a novel approach, called Polygonal Approximation Learning (PAL), is proposed for convex object instance segmentation with only bounding-box supervision. The key idea behind PAL is that the detection model for convex objects already contains the necessary information for segmenting them since their convex hulls, which can be generated approximately by the intersection of bounding boxes, are equivalent to the masks representing the objects. To extract the essential information from the detection model, a repeated detection approach is employed on biomedical images where various rotation angles are applied and a dice loss with the projection of the rotated detection results is utilized as a supervised signal in training our segmentation model. In biomedical imaging tasks involving convex objects, such as nuclei instance segmentation, PAL outperforms the known models (e.g., BoxInst) that rely solely on box supervision. Furthermore, PAL achieves comparable performance with mask-supervised models including Mask R-CNN and Cascade Mask R-CNN. Interestingly, PAL also demonstrates remarkable performance on non-convex object instance segmentation tasks, for example, surgical instrument and organ instance segmentation. Our code is available at https://github.com/shenmishajing/PAL.