Task-oriented grasping of unfamiliar objects is a necessary skill for robots in dynamic in-home environments. Inspired by the human capability to grasp such objects through intuition about their shape and structure, we present a novel zero-shot task-oriented grasping method leveraging a geometric decomposition of the target object into simple, convex shapes that we represent in a graph structure, including geometric attributes and spatial relationships. Our approach employs minimal essential information - the object's name and the intended task - to facilitate zero-shot task-oriented grasping. We utilize the commonsense reasoning capabilities of large language models to dynamically assign semantic meaning to each decomposed part and subsequently reason over the utility of each part for the intended task. Through extensive experiments on a real-world robotics platform, we demonstrate that our grasping approach's decomposition and reasoning pipeline is capable of selecting the correct part in 92% of the cases and successfully grasping the object in 82% of the tasks we evaluate. Additional videos, experiments, code, and data are available on our project website: https://shapegrasp.github.io/.

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