Deep learning has seen remarkable advancements in machine learning, yet it often demands extensive annotated data. Tasks like 3D semantic segmentation impose a substantial annotation burden, especially in domains like medicine, where expert annotations drive up the cost. Active learning (AL) holds great potential to alleviate this annotation burden in 3D medical segmentation. The majority of existing AL methods, however, are not tailored to the medical domain. While weakly-supervised methods have been explored to reduce annotation burden, the fusion of AL with weak supervision remains unexplored, despite its potential to significantly reduce annotation costs. Additionally, there is little focus on slice-based AL for 3D segmentation, which can also significantly reduce costs in comparison to conventional volume-based AL. This paper introduces a novel metric learning method for Coreset to perform slice-based active learning in 3D medical segmentation. By merging contrastive learning with inherent data groupings in medical imaging, we learn a metric that emphasizes the relevant differences in samples for training 3D medical segmentation models. We perform comprehensive evaluations using both weak and full annotations across four datasets (medical and non-medical). Our findings demonstrate that our approach surpasses existing active learning techniques on both weak and full annotations and obtains superior performance with low-annotation budgets which is crucial in medical imaging. Source code for this project is available in the supplementary materials and on GitHub: https://github.com/arvindmvepa/al-seg.