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Spatial perception, particularly at small and medium scales, is an essential human sense but poses a significant challenge for the blind and visually impaired (BVI). Traditional learning methods for BVI individuals are often constrained by the limited availability of suitable learning environments and high associated costs. To tackle these barriers, we conducted comprehensive studies to delve into the real-world challenges faced by the BVI community. We have identified several key factors hindering their spatial perception, including the high social cost of seeking assistance, inefficient methods of information intake, cognitive and behavioral disconnects, and a lack of opportunities for hands-on exploration. As a result, we developed AngleSizer, an innovative teaching assistant that leverages smartphone technology. AngleSizer is designed to enable BVI individuals to use natural interaction gestures to try, feel, understand, and learn about sizes and angles effectively. This tool incorporates dual vibration-audio feedback, carefully crafted teaching processes, and specialized learning modules to enhance the learning experience. Extensive user experiments validated its efficacy and applicability with diverse abilities and visual conditions. Ultimately, our research not only expands the understanding of BVI behavioral patterns but also greatly improves their spatial perception capabilities, in a way that is both cost-effective and allows for independent learning., Comment: The paper was accepted by IMWUT/Ubicomp 2024